diff --git "a/LNFAT4oBgHgl3EQfwR78/content/tmp_files/load_file.txt" "b/LNFAT4oBgHgl3EQfwR78/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/LNFAT4oBgHgl3EQfwR78/content/tmp_files/load_file.txt" @@ -0,0 +1,3086 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf,len=3085 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='08681v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='RT] 20 Jan 2023 A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We initiate a new approach to maximal green sequences by con- sidering them up to an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This reveals extra structure, since the set of equivalence classes of maximal green sequences of an algebra carries interesting partial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We show that the equivalence relation may be defined in several equivalent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We likewise define three conjecturally equivalent partial orders on the set of equivalence classes, and prove some of the implications between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the case of Nakayama algebras we prove that these three partial orders indeed coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partially ordered sets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative projectives in torsion classes: τ-tilting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative simples in torsion classes: bricks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equivalence relations on maximal green sequences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Preliminary lemmas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Polygons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across squares .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Characterising equivalent maximal green sequences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partial orders on equivalence classes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across oriented polygons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partial orders .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Exchange pairs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An example from the twice-punctured torus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Nakayama algebras .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equivalence using bricks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partial order using bricks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 62 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Primary: 16G20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Secondary: 13F60, 16G10, 18E40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences, τ-tilting, silting, torsion classes, Harder– Narasimhan filtrations, cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 1 2 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Bricks versus summands .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 64 References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Introduction Maximal green sequences were introduced by Keller in [Kel11], but were al- ready implicit in the physics literature in the context of BPS spectra of particles in string theory [CCV11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' see also [GMN13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Ali+14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Xie16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The origin of maxi- mal green sequences lies in the theory of cluster algebras, which were introduced by Fomin and Zelevinsky [FZ02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Cluster algebras are commutative rings with distinguished generating sets known as ‘clusters’ which are related by a process called ‘mutation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Every cluster has a skew-symmetrisable matrix associated to it, with mutation both transforming the matrix and the variables of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If the matrix is in fact skew-symmetric, it gives a quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The clusters of a clus- ter algebra form the vertices of a graph whose edges connect clusters related by mutation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' this graph is known as the ‘exchange graph’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Subsequent to their introduction, deep connections were found between cluster algebras and the representation theory of finite-dimensional algebras [MRZ03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' CCS06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Bua+06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here, the clusters of a cluster algebra correspond to objects in a certain category, thereby “categorifying” the cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Some cate- gorifications of cluster algebras — in particular, those related to τ-tilting theory [AIR14] — give an orientation to the edges of the exchange graph, thereby pro- ducing a partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For a helpful survey of this phenomenon, see [BY13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A maximal green sequence is then simply a maximal chain of finite length in the resulting poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The name comes from a combinatorial construction due to Keller [Kel11], in which some vertices of the quiver of a cluster are green and some are red, with a maximal green sequence given by a sequence of mutations at green vertices which turns the quiver from all green to all red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The original motivation for introducing maximal green sequence comes from the theory of Donaldson–Thomas invariants, which, roughly speaking, are count- ing functions of aforementioned BPS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A maximal green sequence for a quiver gives an explicit formula for the refined Donaldson–Thomas invariant as- sociated to the quiver by Kontsevich and Soibelman [KS08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consequently, max- imal green sequences give quantum dilogarithm identities [FV93;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' FK94;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Rei10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The existence of a maximal green sequence also gives a formula for the twist au- tomorphism of a cluster algebra [GLS12], as well as guaranteeing the existence of a theta basis [Gro+18] and a generic basis [Qin22] in the upper cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences are intimately related to Bridgeland stability condi- tions on module categories [Bri07], where a chamber of type I with finitely many A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 3 isomorphism classes of stable objects gives a maximal green sequence [Bri07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A useful survey of maximal green sequences is given in [KD20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It was shown in [Wil22b] that the set of maximal green sequences of an al- gebra could be given more structure by subjecting it to an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, the set of equivalence classes forms a partially ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Posets of maximal green sequences were also studied in [Gor14a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Gor14b] using the lan- guage of [BKT14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Considering maximal green sequences modulo an equivalence relation leads to nice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equivalence classes of maximal green sequences of linearly oriented An are in bijection with triangulations of the three-dimensional cyclic polytope with n + 3 vertices [Wil22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, the partial order on the equivalence classes of these maximal green sequences is a lattice and coincides with the higher Stasheff–Tamari order [Wil22b], which is a higher-dimensional version of the Tamari lattice [KV91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' ER96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In [Wil22b], only one equivalence relation on maximal green sequences is con- sidered and the idea of equivalence of maximal green sequences is not studied as a subject in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, there are several other appealing ways of defining equivalence relations on maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The first main theorem of this paper shows that the equivalence relation on maximal green se- quences admits the following six equivalent characterisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For the detail on the terminology in this theorem, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem A (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be a finite- dimensional algebra over a field K with G and G′ maximal green sequences of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) G and G′ can be deformed into each other across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) G and G′ have the same set of exchange pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) G and G′ have the same set of indecomposable direct summands of two- term silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (4) G and G′ have the same set of indecomposable direct summands of support τ-tilting modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (5) For any Λ-module M, the set of semistable factors of M is the same for the respective Harder–Narasimhan filtrations given by G and G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (6) For any Λ-module M, the multiset of stable factors of M is the same for the respective Harder–Narasimhan filtrations given by G and G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, if G and G′ satisfy any of the equivalent statements above, then G and G′ have the same set of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' However, G and G′ may have the same set of bricks without any of the above holding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The fact that equivalence of maximal green sequences admits several different characterisations indicates that the notion is a natural one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' On the other hand, it is rather surprising that two maximal green sequences may have the same set 4 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS of bricks while having different sets of τ-rigid modules, since bricks are dual to indecomposable τ-rigid modules [DIJ19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is related to the “tropical duality” that exists between c-vectors and g-vectors for cluster algebras [NZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' However, we show that two maximal green sequences with the same bricks are equivalent in the case of Nakayama algebras (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' More generally, as we explain, having the same factors of Harder–Narasimhan filtrations can be considered an augmentation of the condition of having the same bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An intriguing impli- cation of Theorem A is that the set of indecomposable summands of support τ-tilting modules of a maximal green sequence determines the semistable fac- tors of every module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' it would be interesting if there were a direct construction relating the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Having introduced equivalence relations on maximal green sequences, we can define partial orders on the equivalence classes, as again in [Wil22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, considering maximal green sequences subject to an equivalence relation reveals extra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' These partial orders can also be seen in the context of partial orders on chambers of stability conditions, although this is not the language we choose to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' There is a partial order on the chambers of King stability conditions [Kin94] given by those from τ-tilting theory [BST19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Asa21], and a partial order on chambers of type II for Bridgeland stability conditions given by inclusion of aisles of t-structures [Bri07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Roughly speaking, we study partial orders on equivalence classes of certain type I chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' As an aside, it is curious to note that while all two-term silting complexes give chambers of King stability conditions [Bri17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' BST19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Asa21], not all maximal green sequences give chambers of Bridgeland stability conditions, see [Qiu15, Counterexample 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='16] and [AI20, Theorem L3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We introduce three partial orders on equivalence classes of maximal green sequences: one in terms of certain deformations, one in terms of indecompos- able summands of two-term silting complexes, and one in terms of Harder– Narasimhan filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We conjecture that these partial orders coincide, to- wards which we show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1 for the definition of an increasing elementary polygonal deformation and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5(3) for the definition of refinement of Harder–Narasimhan filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem B (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='14 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be a finite-dimensional algebra over a field K with G and G′ maximal green sequences of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Further- more, suppose that [G′] is the result of a series of increasing elementary polygonal deformations of [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) Every indecomposable direct summand of a two-term silting complex of G′ is a direct summand of a two-term silting complex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) The Harder–Narasimhan filtrations induced by G′ refine those induced by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 5 The partial order given by deformations is very natural in terms of stability conditions, since increasing elementary polygonal deformations correspond to crossing walls of type I which decrease the number of stables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We show that in certain cases these orders have unique maxima and minima (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For Nakayama algebras, or algebras with two simple modules up to isomor- phism, we prove the converse implications from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Namely, we show that the three partial orders in fact coincide, along with an additional order given by inclusion of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The latter is not well-defined for an arbitrary finite- dimensional algebra Λ, since in general non-equivalent maximal green sequences may have the same bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem C (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='19 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be a finite-dimensional algebra over a field K which is either a Nakayama algebra or an algebra with two isomorphism classes of simple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Further, let G and G′ maximal green sequences of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) [G′] is the result of a series of increasing elementary polygonal deforma- tions of [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) Every indecomposable direct summand of a support τ-tilting module of G′ is a direct summand of a support τ-tilting module of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) The Harder–Narasimhan filtrations induced by G′ refine those induced by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (4) Every brick of G′ is a brick of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Organisation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We begin in Section 2 by giving background to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We introduce maximal green sequences in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2, and then give their description in terms of τ-tilting theory in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3, followed by their description in terms of sequences of bricks in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We introduce equivalence relations on maximal green sequences in Section 3, and prove our first main result Theorem A showing the different ways of characterising the equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In Section 4, we introduce three partial orders on equivalence classes of maximal green sequences and prove our second main result Theorem B, showing how these are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then go on to consider the interaction between one of the partial orders and the exchange pairs of a maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then discuss an interesting example of a poset of equivalence classes of maximal green sequences of an algebra related to a triangulation of the twice-punctured torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Finally, in Section 5 we study the posets for Nakayama algebras in detail and prove Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 6 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We would like to thank Aran Tattar for help with the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='16, and Haruhisa Enomoto, Bernhard Keller, Hipolito Treffin- ger, Osamu Iyama, Aaron Chan, Daniel Labardini-Fragoso, and H˚avard Terland for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This work is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Hori- zon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 101001159).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Parts of this work were done during stays of MG at the University of Stuttgart, and he is very grateful to Steffen Koenig for the hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' NJW is currently supported by EPSRC grant EP/V050524/1, and part of the work on this paper was done while a JSPS short-term postdoctoral research fellow at the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Throughout this paper, we let Λ be a finite-dimensional algebra over a field K, with mod Λ the category of finitely generated right Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this paper we use the symbols ‘⊂’ and ‘⊃’ to denote strict inclusion of sets, that is, inclusion but not equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is more commonly denoted with the symbols ‘⊊’ and ‘⊋’ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a set P, a partial order on P is a relation R ⊆ P × P which is reflexive, symmetric, and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partially ordered sets are referred to as posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We usually write partial orders with the symbol ⩽, so that if (x, y) ∈ R, where R is a partial order on a set P, we write x ⩽ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A covering relation in a partial order ⩽ is a relation x < z such that if x ⩽ y ⩽ z, then y = x or y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One can also say that z covers x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is usual to write x ⋖ z when x < z is a covering relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An interval of a poset P is a subset of the form {y ∈ P | x ⩽ y ⩽ z} for some x, z ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The Hasse diagram of a partial order ⩽ on a set P is the quiver with the elements of P as vertices, with arrows z → x whenever x ⋖ z is a covering relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this paper, we illustrate posets using their Hasse diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Recall that a Hasse diagram is n-regular if every vertex is incident to precisely n arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences were introduced by Keller in the context of Donaldson–Thomas theory using a combinatorial definition in terms of quivers [Kel11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It follows from work of Nagao that this is equivalent to having a maximal chain of torsion classes [Nag13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is the first notion of a maximal green sequence that we will cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Torsion pairs were introduced by Dickson to generalise the structure given by torsion and torsion-free abelian groups to arbitrary abelian categories [Dic66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A torsion pair is a pair of full subcategories (T , F) of mod Λ such that A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 7 (1) HomΛ(T , F) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) if HomΛ(T, F) = 0, then T ∈ T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) if HomΛ(T , F) = 0, then F ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here T is called the torsion class and F is called the torsion-free class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' More generally, a full subcategory T is called a torsion class if it is a torsion class in some torsion pair, and likewise for torsion-free classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is well-known that a full subcategory T of mod Λ is a torsion class if and only if it is closed under factor modules and extensions [Dic66, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a set X of Λ-modules, we write Tors(X) for the smallest torsion class containing X and Torf(X) for the smallest torsion-free class containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a torsion pair (T , F) in mod Λ and a Λ-module M, there is an exact sequence 0 → L → M → N → 0 such that L ∈ T and N ∈ F, which is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here L is called the torsion submodule of M and N is called the torsion-free factor module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The torsion classes of mod Λ form a complete lattice under inclusion, denoted tors Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We call the covering relations of this lattice minimal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, T ⊂ T ′ is a minimal inclusion if and only if whenever we have T ⊆ T ′′ ⊆ T ′, we must either have T ′′ = T or T ′′ = T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will be particularly interested in the subposet ff-tors Λ of functorially finite torsion classes of Λ, where ‘functorially finite’ is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a subcategory X ⊆ mod Λ and a map f : X → M, where X ∈ X and M ∈ mod Λ, we say that f is a right X-approximation if for any X′ ∈ X, the sequence HomΛ(X′, X) → HomΛ(X′, M) → 0 is exact, following [AS80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Left X-approximations are defined dually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The sub- category X is said to be contravariantly finite if every M ∈ mod Λ admits a right X-approximation, and covariantly finite if every M ∈ mod Λ admits a left X-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If X is both contravariantly finite and covariantly finite, then X is functorially finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Certain sorts of approximations are of particular note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A morphism f : X → Y is right minimal if any morphism g: X → X such that fg = f is an isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Left minimal morphisms are defined dually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A right X-approximation is a minimal right X-approximation if it is also right minimal, and minimal left approximations are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' First notion of maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A maximal green sequence is a maximal chain in tors Λ of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' More explicitly, a maximal green se- quence is a chain of minimal inclusions of torsion classes mod Λ = T0 ⊃ T1 ⊃ · · · ⊃ Tr−1 ⊃ Tr = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 8 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS This is the first definition of a maximal green sequence that we shall see, but we shall see two further notions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We shall regard the three notions as being cryptomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We note at this point that a result of Demonet, Iyama, and Jasso [DIJ19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1] implies that every torsion class in a finite maximal chain is func- torially finite, so that maximal green sequences are in fact finite maximal chains in ff-tors Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative projectives in torsion classes: τ-tilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Our second notion of maximal green sequences will operate in terms of the relative projectives of the torsion classes in the maximal chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative projectives in torsion classes were studied by Adachi, Iyama, and Reiten in [AIR14] in terms of what is called ‘τ-tilting theory’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative projectives in torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a torsion class T ∈ tors Λ, a module X ∈ T is a relative projective if Ext1 Λ(X, M) = 0 for all M ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We write P(T ) for the direct sum of one copy of each indecomposable relative projective in T , up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Support τ-tilting pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative projectives in torsion classes correspond to what are called ‘support τ-tilting modules’, where τ denotes the Auslander– Reiten translate in mod Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A Λ-module M is called τ-rigid if HomΛ(M, τM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A pair (M, P) of Λ-modules where P is projective is called τ-rigid if M is τ- rigid and HomΛ(P, M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A τ-rigid pair (M, P) is called support τ-tilting if |M|+|P| = |Λ|, where |X| denotes the number of non-isomorphic indecomposable direct summands of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this case M is called a support τ-tilting module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We write sτ-tilt Λ for the set of isomorphism-class representatives of basic support τ-tilting modules over Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1 ([AIR14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' There is a bijection ff-tors Λ ←→ sτ-tilt Λ, T �−→ P(T ), Fac M ←−� M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here Fac M := {X ∈ mod Λ | ∃ an epimorphism M⊕m ։ X for some m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Two-term silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An equivalent framework to support τ-tilting modules is given by two-term silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will often prefer to work with these objects instead, for reasons that we will explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We denote by Kb(proj Λ) the homotopy category of bounded complexes of projective right Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will often want to consider K[−1,0](proj Λ), the A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 9 subcategory of Kb(proj Λ) consisting of two-term complexes, that is, complexes concentrated in degrees −1 and 0: P −1 → P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An object T of Kb(proj Λ) is pre-silting if HomKb(proj Λ)(T, T[i]) = 0 for all i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A pre-silting complex T is silting if, additionally, thick T = Kb(proj Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here thick T denotes the smallest full subcategory of Kb(proj Λ) which contains P and is closed under cones, [±1], direct summands, and isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For a two-term complex T to be pre-silting, it suffices that HomKb(proj Λ)(T, T[1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, for a pre-silting two-term complex T to be silting, it suffices that |T| = |Λ| by [AIR14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We write 2-silt Λ for the set of isomorphism-class representatives of basic two-term silting complexes of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2 ([AIR14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' There is a bijection 2-silt Λ ←→ sτ-tilt Λ, T �−→ H0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, support τ-tilting and two-term silting are essentially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Func- torially finite torsion classes of Λ are therefore also in bijection with two-term silting complexes over Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The advantage of support τ-tilting is that it is easier to describe the relation with torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The advantage of two-term silting is that it is easier to talk about mutation, which is why we often work in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a silting complex T = E ⊕ X in Kb(proj Λ) where X is indecomposable, let X f−→ E′ g−→ Y → X[1] be a triangle in Kb(proj Λ) such that f is a minimal left add E-approximation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This triangle is known as the exchange triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, by [AI12, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='35], Y is indecomposable with g a minimal right add E-approximation of Y and, by [AI12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='31], T ′ = E ⊕ Y is a silting complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this situation, we say that T ′ is a green mutation (or left mutation) of T and T is a red mutation of T ′ (or right mutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The opposite convention for green and red mutations is used by some authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Such choice would not make any differ- ence to our considerations of maximal chains in the lattice of two-term silting complexes, since the chain remains the same whichever direction one traverses it in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We call (X, Y ) the exchange pair of the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We say that a pair of two-term silting complexes T, T ′ ∈ K[−1,0](proj Λ) are mutations of each other if and only if T = E ⊕ X and T ′ = E ⊕ Y where X and Y are indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By [AIR14, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8(b)], we have that T and T ′ are mutations of each other if and only if either T ′ is a green mutation of T, 10 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS or T is a green mutation of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The set of basic two-term silting complexes of Λ forms a poset denoted 2-silt Λ where the covering relations are that T ′ ⋖ T if and only if T ′ is a green mutation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The partial order itself is then the transitive–reflexive closure of these covering relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3 ([AIR14, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='34, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The bijection between 2-silt Λ and ff-tors Λ induces an isomorphism between the Hasse diagrams of these posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In particular, if functorially finite torsion classes T and T ′ correspond to two- term silting complexes T and T ′ respectively, then there is a minimal inclusion T ⊃ T ′ if and only if T ′ is a green mutation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Second notion of maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We can use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3 to obtain the second notion of maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since a maximal green sequence is a maximal chain of minimal inclusions in ff-tors Λ, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3, we have that a maximal green sequence is a maximal sequence of green mutations of two-term silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' More explicitly, a maximal green sequence is a sequence of two-term silting complexes Λ = T0, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Tr = Λ[1] such that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , r}, we have that Ti is a green mutation of Ti−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This was first observed by Br¨ustle, Smith, and Treffinger [BST19, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Such a maximal green sequence can be specified by labelling each of the sum- mands of Λ from 1 to n where n = |Λ| and then giving a list of numbers specifying the sequence of summands to be mutated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (When summand i is mutated, the summand that replaces it is then also labelled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=') In terms of the original notion of maximal green sequence from [Kel11] using quiver mutation, this can be seen as a sequence of vertices of the quiver to mutate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will use this perspective a couple of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative simples in torsion classes: bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now give background leading up to the third notion of maximal green sequences, which comes from looking at the relative simples in torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relative simples in torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Relatively simple objects in exact cat- egories were first studied in [BG16] and were considered in the specific case of torsion-free classes in [Eno21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A Λ-module B in a torsion class T is a relative simple if there is no short exact sequence 0 → A → B → C → 0 such that A and C are both non-zero modules in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since torsion classes are closed under factor modules, it is in fact necessary and sufficient for B to have no proper submodules which are in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We write simp(T ) for the set of isomorphism- class representatives of relative simples of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Brick labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In a somewhat, but not entirely, analogous way to how rel- ative projectives of torsion classes correspond to support τ-tilting modules, rel- ative simples of torsion classes correspond to certain modules known as ‘bricks’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An object B of mod Λ is a brick if EndΛ B is a division ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equivalently, B is a brick if every non-zero endomorphism of B is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is the case if and only if B has no proper factor module which is isomorphic to a proper submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4 ([BCZ19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Dem+18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have that T ⊇ U is a minimal inclusion if and only if T ∩ U⊥0 = Filt(B) for a brick B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, this brick B is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here Filt and U⊥0 are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have U⊥0 := {M ∈ mod Λ | HomΛ(U, M) = 0 for all U ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a full subcategory C of mod Λ, Filt(C) is the full subcategory of mod Λ consisting of modules M with a finite filtration M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0 such that Mi−1/Mi ∈ C for all 1 ⩽ i ⩽ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is known that Filt(B) for a brick B is a wide subcategory of mod Λ [Rin76, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2], meaning that it is closed under extensions, kernels and cokernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is therefore also closed under images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this way, the covering relations of tors Λ can be labelled by bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We often write the brick labels of the inclusions by T B⊃ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For two torsion classes T ⊇ U, we denote [T , U] := T ∩U⊥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The relation between brick labels and intervals in the lattice of torsion classes extends beyond intervals given by covering relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5 ([Tat21, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8], [Eno21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Ti Bi+1 ⊃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Bj⊃ Tj be a chain of minimal inclusions in tors Λ with brick labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then [Ti, Tj] = Filt(Bi+1, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The following proposition is a slight generalisation of [Eno21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given torsion classes Ti ⊇ Tj in mod Λ, every relatively sim- ple object in [Ti, Tj] must occur as a brick label in every finite maximal chain in the interval of tors Λ between Ti and Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 12 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' To see this, take a relatively simple object S in [Ti, Tj] and sup- pose that there is a maximal chain connecting Ti and Tj with brick labels Bi+1, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, since [Ti, Tj] = Filt(Bi+1, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bj), we must have that S has a filtration with factors in {Bi+1, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But, since S is relatively simple in [Ti, Tj], this filtration can only have one factor, and so we must have that S = Bk for some k, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ Relatively simple objects in [Ti, Tj] were studied in [AP22] in the case where this category is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Finally, the relationship between the relative simples of torsion classes in a maximal green sequence and the brick labels is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='7 ([Eno21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence mod Λ = T0 B1 ⊃ T1 B2 ⊃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Br−1 ⊃ Tr−1 Br ⊃ Tr = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then r� i=0 simp(Ti) = {B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' More precisely, [Eno21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8] proves the inclusion r� i=0 simp(Ti) ⊆ {B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br}, and the converse inclusion follows from the fact that Bi is a relative simple in Ti−1, for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Third notion of maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We can now give the third notion of maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the appendix to [KD20], Demonet shows that (not necessarily finite) maximal chains of torsion classes may be characterised in terms of bricks, generalising [Igu19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Another related result to this is [Tre20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A backwards Hom-orthogonal sequence of bricks is a sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br such that if i < j then HomΛ(Bj, Bi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A backwards Hom-orthogonal se- quence of bricks is maximal if one cannot insert a brick at any point in the sequence without losing the backwards Hom-orthogonality property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By [KD20, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3], there is a bijection between maximal green sequences and max- imal backwards Hom-orthogonal sequences of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a maximal green sequence mod Λ = T0 ⊃ T1 ⊃ · · · ⊃ Tr = {0}, one obtains the maximal backwards Hom-orthogonal sequence of bricks by taking Bi as the brick label of the minimal inclusion Ti−1 ⊃ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Conversely, given a A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 13 maximal backwards Hom-orthogonal sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br, one obtains the corresponding maximal green sequence by taking Ti to be the smallest torsion class Tors(Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br) containing Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Harder–Narasimhan filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It was shown in [Igu20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8] that maximal green sequences induce so-called “Harder–Narasimhan filtra- tions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This was then shown for arbitrary chains of torsion classes in [Tre20, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='10], see also [BKT14, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Such filtrations were originally studied in [HN75] and are well known from their role in the theory of stability conditions, for example [Rud97;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Bri07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br be a maximal green sequence G of a finite-dimensional algebra Λ given as a maximal backwards Hom-orthogonal sequence of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, every non-zero Λ-module M has a unique filtration M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0 such that Fj := Mj−1/Mj ∈ Filt(Bij) for some ij, with i1 < i2 < · · · < il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This filtration is called the Harder–Narasimhan filtration (HN filtration) or the G-Harder–Narasimhan filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is precisely the filtration of M by the torsion submodules associated to it by the torsion classes in G, with duplicated torsion submodules removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We call Fj the semistable factors here, or the G-semistable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We write SSFG(M) = {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Fl} for the set of G-semistable factors of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since the module Fj lies in Filt(Bij), it has a filtration where all of the factors are isomorphic to Bij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (In terms of stability conditions, this is the Jordan–H¨older filtration of a semistable module in terms of stable modules [Rud97, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=') We write SFG(Fj) for the multiset of Bij factors in this filtration of Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' That is, SFG(Fj) is s copies of Bij, where s is the number of factors in this filtration of Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We furthermore write SFG(M) = l� i=1 SFG(Fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We refer to SFG(M) as the multiset of stable factors or G-stable factors of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 14 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equivalence relations on maximal green sequences In this section, we define equivalence of maximal green sequences and give six different criteria for a pair of maximal green sequences to be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We also show that two non-equivalent maximal green sequences can have the same sets of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, having the same set of bricks is not a sufficient criterion for two maximal green sequences to be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Before we start characterising equivalent maximal green sequences, we must prove some preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the following lemma, we collect some useful facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here, for T ∈ K[−1,0](proj Λ), we denote T ⊥1 = {X ∈ K[−1,0](proj Λ) | HomKb(proj Λ)(T, X[1]) = 0}, ⊥1T = {X ∈ K[−1,0](proj Λ) | HomKb(proj Λ)(X, T[1]) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let T = U ⊕ X and T ′ = U ⊕ Y be two two-term silting complexes in K[−1,0](proj Λ) such that T ′ is a green mutation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then: (1) ⊥1T ⊂ ⊥1T ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' T ⊥1 ⊃ T ′⊥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) X ∈ ⊥1T ′, Y ∈ T ⊥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) X /∈ T ′⊥1, Y /∈ ⊥1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) follows from [AI12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) then follows immediately from this, since clearly X ∈ ⊥1T and Y ∈ T ′⊥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) then follows, since, by assumption, T ′ ⊕ X and T ⊕ Y cannot be silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ By iterating Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence of Λ containing two-term silting complexes T and T ′ where T ′ occurs after T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Furthermore, let X be an indecomposable summand of T which is not a summand of T ′ and let Y be an indecomposable summand of T ′ which is not a summand of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then: (1) ⊥1T ⊂ ⊥1T ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' T ⊥1 ⊃ T ′⊥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) X ∈ ⊥1T ′, Y ∈ T ⊥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) X /∈ T ′⊥1, Y /∈ ⊥1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We consider G to be a sequence T0, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Tr of green mutations of two-term silting complexes with T0 = Λ and Tr = Λ[1] in K[−1,0](proj Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) We denote by Ss(G) := r� i=0 {Indecomposable summands of Ti}/ ∼= the set of isomorphism classes of indecomposable complexes which occur as direct summands of two-term silting complexes in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 15 (2) We denote by E(G) := r� i=1 {(X, Y ) | Ti−1 = E ⊕ X, Ti = E ⊕ Y } the set of exchange pairs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Furthermore, now consider G to be a sequence of support τ-tilting modules M0, M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Mr corresponding to the two-term silting complexes Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) We denote by Sτ(G) := r� i=0 {Indecomposable summands of Mi}/ ∼= the set of isomorphism classes of indecomposable modules which occur as direct summands of support τ-tilting modules in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Finally, consider G to be a maximal backwards Hom-orthogonal sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (4) We denote the set of bricks in the sequence by B(G) := {B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The following fact is well-known, but we do not know whether a proof has ap- peared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We use ind K[−1,0](proj Λ) to denote a set of representa- tives of the isomorphism classes of indecomposable complexes in K[−1,0](proj Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let A ∈ ind K[−1,0](proj Λ) and let G be a maximal green sequence of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then there is at most one exchange pair (X, Y ) ∈ E(G) such that X ∼= A and at most one exchange pair (X, Y ) ∈ E(G) such that Y ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, the exchange pair (X, A) must occur before the exchange pair (A, Y ) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose for contradiction that G possesses two exchange pairs (A, X1) and (A, X2) which respectively correspond to green mutations from T1 to T ′ 1 and from T2 to T ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose without loss of generality that T1 occurs before T2 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2, we obtain that A /∈ T ′ 1 ⊥1 ⊇ T2⊥1 ∋ A, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The case where A is the second half of the exchange pair is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For the final statement it suffices to observe that if (A, Y ) occurs without (X, A) before it, then A must be a projective, and if (X, A) occurs without (A, Y ) after it, then A must be a shifted projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But both cannot simultaneously be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ We finish this subsection by proving the following useful result on exchange pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Exchange pairs have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 16 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS (1) Suppose that (X, Y ) is an exchange pair for a green mutation from T to T ′ in a maximal green sequence G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then X is the unique indecomposable in Ss(G) ∩ T ⊥1 such that HomKb(proj Λ)(Y, X[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) Conversely, let T be a two-term silting complex in a maximal green se- quence G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that, for Y ∈ Ss(G) ∩ T ⊥1, there exists an indecom- posable X ∈ Ss(G) ∩ T ⊥1, unique up to isomorphism, such that HomKb(proj Λ)(Y, X[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then (X, Y ) ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) Suppose that there exists an indecomposable two-term complex Z ∈ Ss(G) ∩ T ⊥1 such that HomKb(proj Λ)(Y, Z[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then Z /∈ T ′⊥1, since Y is a summand of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2, we cannot have that Z is a summand of T ′ or any silting complex which occurs later in G than T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If Z leaves G before T, then Z /∈ T ⊥1, which is contrary to our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We conclude that Z must leave G between T and T ′, and so Z ∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) Now suppose that (X, Y ) is such that Y ∈ Ss(G) ∩ T ⊥1 and X is the unique indecomposable in Ss(G) ∩ T ⊥1 such that HomKb(proj Λ)(Y, X[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The fact that X ∈ T ⊥1 means that we cannot have Y as a summand of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Therefore, Y cannot occur in G before T by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2(3), otherwise Y /∈ T ⊥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Thus, Y occurs in G after T as the second half of an exchange pair (Z, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But then we must have that Z ∈ Ss(G) ∩ T ⊥1 by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2(2) and that HomKb(proj Λ)(Y, Z[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We obtain that Z ∼= X, since X is the unique such indecomposable up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One of the equivalence relations we introduce on maximal green sequences corresponds to deformations across squares in 2-silt Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Squares are a special case of the larger class of “polygons” in 2-silt Λ, so we introduce all polygons at this juncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Several authors have studied the notion of a polygon in the poset of torsion classes or two-term silting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A lattice-theoretic notion is used by Reading [Rea16], and Garver and McConville [GM19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Our notion is instead based on that of Hermes and Igusa [HI19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A polygon in the poset 2-silt Λ is a subposet consisting of all two-term silting complexes possessing some presilting complex E as a direct summand, where |E| = |Λ| − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A polygon in ff-tors Λ is the image of a polygon in 2-silt Λ under the bijection between the two posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A polygon in the poset 2-silt Λ falls under one of the four different types shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The two-regularity of the Hasse diagram of the polygon follows from the fact that all but two summands are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The existence of a unique maximum and minimum follows from the existence of the Bongartz and co-Bongartz com- pletions [AIR14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The cases displayed are then clearly exhaustive, which are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (a) Square: the two paths from the maximum to the minimum are both of length two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (b) Oriented polygon: one path from the maximum to the minimum is of length two and the other is finite of length greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (c) Unoriented polygon: both paths from the maximum to the minimum are finite of length greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (d) Infinite polygon: the polygon contains infinitely many elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Equiv- alently, there is at most one path of finite length from the maximum to the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A finite polygon in 2-silt Λ is a polygon which is not an infinite polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A finite polygon therefore has two finite paths from its maximum to its minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Two maximal green sequences G and G′ are related by deformation across a (finite) polygon if we have G = (T0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Ti, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Uk, Ti+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Tr), G′ = (T0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Ti, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Vl, Ti+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Tr), as sequences of green mutations of silting complexes, with Ti U1 Uk Ti+1 V1 Vl 18 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Different types of polygons (a) Square (b) Oriented polygon 2 < length < ∞ (c) Unoriented polygon 2 < length < ∞ 2 < length < ∞ (d) Infinite polygon length = ∞ a finite polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the case of a finite polygon, the two paths around the polygon from the maximum to the minimum are the only paths between these silting complexes in the poset 2-silt Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let E ⊕ X ⊕ X′ and E ⊕ Y ⊕ Y ′ be the respective maximum and minimum of a polygon in 2-silt Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then the only paths from E ⊕ X ⊕ X′ to E ⊕ Y ⊕ Y ′ in 2-silt Λ are the two paths around the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Each vertex in the Hasse diagram of the polygon has degree two, corre- sponding to the two indecomposable summands of the two-term silting complex at the vertex which are not summands of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Having another path from E⊕X⊕X′ to E ⊕ Y ⊕ Y ′ would require mutating an indecomposable direct summand of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' However, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4 then precludes the path reaching E ⊕ Y ⊕ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 19 Any convex subposet which looks like a square or an oriented polygon must in fact be a square or an oriented polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Recall that a subposet P′ of a poset P is convex if whenever p ⩽ q ⩽ r with p, r ∈ P′, we have that q ∈ P′ too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Any convex subposet of 2-silt Λ isomorphic to Figure 1(a) or 1(b) is a polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This follows from the fact that in each of these cases there is a path from the maximum of the poset to the minimum of the poset of length two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since the covering relations from Figure 1 correspond to covering relations in 2-silt Λ, we must have that the maximum of the poset is two mutations away from the bottom of the poset, and so shares all but two summands with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, there exists E such that |E| = |Λ| − 2 such that E is a summand of all two-term silting complexes along the length-two path from maximum to minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It then follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4 that E must be a summand of all the two-term silting complexes along the other path too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We conclude that the subposet is indeed a polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The different types of polygons correspond to the two-term silting theory of different algebras Γ with |Γ| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, let E be a two-term presilting complex with |E| = |Λ| − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then the complexes which complete E to a silting complex must lie in Z = (⊥0E[> 0]) ∩ (E[< 0]⊥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here ⊥0E[> 0] = {X ∈ Kb(proj Λ) | HomKb(proj Λ)(X, E[i]) = 0 ∀i > 0}, E[< 0]⊥0 = {X ∈ Kb(proj Λ) | HomKb(proj Λ)(E[i], X) = 0 ∀i < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let � (−): Z → Z/[E] be the ideal quotient of Z by the ideal of morphisms factoring through add E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then have that Z/[E] is a triangulated category by [IY08, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Further, let E ⊕ X ⊕ Y be the Bongartz completion of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, it follows from [Jas15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' IY18] that the polygon determined by E is isomorphic as a poset to 2-silt Γ where Γ = EndZ/[E](X ⊕ Y ) via sending a two-term silting complex E ⊕ X′ ⊕ Y ′ in K[−1,0](proj Λ) to the two-term silting complex � X′ ⊕ � Y ′ of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This process is known as silting reduction and was introduced in [AI12] and given this description in [IY18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A particular instance of deformation across a polygon is deformation across a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This will be used to define one notion of equivalence for maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this section, we study deformations across squares in terms of silting complexes and bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We first note the following straightforward characterisation of deformations across squares in terms of chains of torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 20 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given maximal green sequences G T0 ⊃ T1 ⊃ · · · ⊃ Tr−1 ⊃ Tr and G′ T ′ 0 ⊃ T ′ 1 ⊃ · · · ⊃ T ′ r′−1 ⊃ T ′ r′ of Λ, we have that G and G′ are related by deformation across a square if and only if r = r′ and there is some j such that Ti = T ′ i for all i ̸= j, and Tj ̸= T ′ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across squares in terms of silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We prove some results on deformations across squares from the perspective of silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If the exchange pairs in one path around the square are (X, Y ) and then (X′, Y ′), then the exchange pairs in the path around the other side of the square are (X′, Y ′) and then (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let T be the two-term silting complex at the top of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then X is certainly an indecomposable summand of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We must also have that X′ is an indecomposable summand of T, since we cannot have X′ ∼= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, let T = E ⊕ X ⊕ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This gives the path around the square we know as E ⊕ X ⊕ X′ → E ⊕ Y ⊕ X′ → E ⊕ Y ⊕ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let T ′ be the two-term silting complex in the middle of the path around the other side of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We must have that T ′ has all but one indecomposable summand in common with E ⊕ X ⊕ X′ and E ⊕ Y ⊕ Y ′, since it is related to each of these silting complexes by mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is then immediate to see that T ′ ∼= E ⊕ X ⊕ Y ′, which gives us the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ The following criterion for when one can swap the order of two exchange pairs will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence of Λ with an exchange pair (X, Y ) immediately succeeded by an exchange pair (X′, Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then one can deform G across a square with sides (X, Y ) and (X′, Y ′) to obtain another maximal green sequence G′ if and only if HomKb(proj Λ)(Y ′, X[1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Ti−1 Ti Ti+1 (X,Y ) (X′,Y ′) be the relevant part of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that we can deform G across a square with these sides to give a maximal green sequence G′, which, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='13, instead has the sequence Ti−1 T ′ i Ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (X′,Y ′) (X,Y ) Then T ′ i contains both Y ′ and X as summands, and so we must have that HomKb(proj Λ)(Y ′, X[1]) = 0, since T ′ i is silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 21 Now we show the reverse direction, maintaining our labelling of G as above, and supposing that HomKb(proj Λ)(Y ′, X[1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then have that Ti−1 = E ⊕ X ⊕ X′ for some E ∈ K[−1,0](proj Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, if we let T ′ i = E ⊕ X ⊕ Y ′, then HomKb(proj Λ)(T ′ i, T ′ i[1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, we have the following: HomKb(proj Λ)(E ⊕ Y ′, (E ⊕ Y ′)[1]) = 0, as E ⊕ Y ′ is a summand of Ti+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' HomKb(proj Λ)(E ⊕ X, (E ⊕ X)[1]) = 0, as E ⊕ X is a summand of Ti−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' HomKb(proj Λ)(Y ′, X[1]) = 0 by assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' HomKb(proj Λ)(X, Y ′[1]) = 0, since Y ′ ∈ Ti⊥1 ⊂ Ti−1⊥1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4, we have Y ′ ̸∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Therefore, T ′ i has the maximal number of isomorphism classes of indecomposable summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence T ′ i is a silting complex and we obtain a maximal green sequence G by replacing the relevant portion of G with Ti−1 T ′ i Ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (X′,Y ′) (X,Y ) □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across squares in terms of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now consider defor- mations across squares from the point of view of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a maximal green sequence G of Λ as the maximal backwards Hom-orthogonal sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br, a maximal green sequence G′ is related to G by deformation across a square if and only if G′ is given by B0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi−1, Bi+1, Bi, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br as a maximal backwards Hom-orthogonal sequence of bricks for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The maximal green sequences G and G′ are related by deformation across a square if and only if there is a square Ti−1 Ti T ′ i Ti+1 such that G and G′ differ only in that G contains the left-hand path around the square, whilst G′ contains the right-hand path around the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Bi and Bi+1 be the respective brick labels of the minimal inclusions Ti−1 ⊃ Ti and Ti ⊃ Ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then we have that [Ti−1, Ti+1] = Filt(Bi, Bi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, Bi and Bi+1 must be precisely the relatively simple objects in Filt(Bi, Bi+1) since if either Bi ∈ Filt(Bi+1) or Bi+1 ∈ Filt(Bi), then backwards Hom-orthogonality 22 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS is violated, and there cannot be more relatively simple objects in Filt(Bi, Bi+1) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6 to the other path around the square, we see that the brick labels of the other two minimal inclusions must also be Bi and Bi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then must have that Bi+1 labels Ti−1 ⊃ T ′ i and Bi labels T ′ i ⊃ Ti+1, since Ti ̸= T ′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ In order to prove the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='14 for bricks, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that L and N are bricks over Λ such that HomΛ(L, N) = HomΛ(N, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then every non-split extension of L and N is a brick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that 0 → L f−→ M g−→ N → 0 is a non-split extension of L and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We want to show that M is a brick, that is, that every non-zero morphism h: M → M is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since HomΛ(L, N) = 0, we have that ghf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, by the universal properties of kernels and cokernels, we have a commutative diagram 0 L M N 0 0 L M N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' f a g h b f g Since L and N are both bricks, we have that a and b are both either isomorphisms or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If they are both isomorphisms, then h is also an isomorphism by the Five Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, we suppose that at least one of a and b is not an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose first that a is zero and b is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Thus, hf = fa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By the universal property of cokernels, we have a map s: N → M such that sg = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 0 L M N 0 0 L M N 0 f 0 g h b s f g We then have that gsg = gh = bg, which implies that gs = b, since g is epic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since b is an isomorphism, it has an inverse b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then have that gsb−1 = id, so that sb−1 is a section of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But this means that the extension 0 → L → M → N → 0 is split, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The case where a is an isomorphism and b is zero is similar to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The final case to consider is where a and b are both zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This gives that hf = fa = 0, and so we have a map s: N → M such that h = sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 23 gsg = gh = bg = 0, which implies that gs = 0, since g is epic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By the universal property of kernels, we have that there is a map t: N → L such that s = ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 0 L M N 0 0 L M N 0 f 0 g h 0 s t f g However, HomΛ(N, L) = 0, so t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consequently, s = ft = 0 and, in turn, h = sg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We conclude that every endomorphism of M is either an isomorphism or zero, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ We apply this to show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence of Λ given by a maximal backwards Hom-orthogonal sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi, Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If HomΛ(Bi, Bi+1) = 0, then Ext1 Λ(Bi+1, Bi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If Ext1 Λ(Bi+1, Bi) ̸= 0, then there is a non-split short exact sequence 0 → Bi → B → Bi+1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The module B here must then be a brick by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We claim that the sequence of bricks given by B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi−1, Bi, B, Bi+1, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br is also backwards Hom-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose there exists Bj with j > i + 1 such that there is a non-zero homomorphism Bj → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The composition Bj → B → Bi+1 must then be zero, by backwards Hom-orthogonality of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But this gives a non-zero map Bj → Bi by the universal property of the kernel, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One can similarly argue that there is no non-zero map B → Bj for j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We must finally show that there cannot be any non-zero maps Bi+1 → B or B → Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the first case, if the composition Bi+1 → B → Bi+1 is zero, then there is a contradictory non-zero map Bi+1 → Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence the composition Bi+1 → B → Bi+1 is non-zero and cannot be an isomorphism since B is inde- composable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This contradicts the fact that Bi+1 is a brick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The existence of a non-zero map B → Bi is likewise contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ The following lemma is the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='14 for bricks: it tells us when we can exchange two consecutive bricks in order to deform across a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 24 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G be a maximal green sequence of Λ given by a maximal backwards Hom-orthogonal sequence of bricks B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi, Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi−1, Bi+1, Bi, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br is a maximal backwards Hom-orthogonal sequence of bricks if and only if HomΛ(Bi, Bi+1) = Ext1 Λ(Bi, Bi+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We first show that the conditions are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is immediate that we must have HomΛ(Bi, Bi+1) = 0 if the sequence is to remain backwards Hom- orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='17, we then also have that Ext1 Λ(Bi, Bi+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now show that the conditions are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have that the new sequence is backwards Hom-orthogonal as HomΛ(Bi, Bi+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose now that the new sequence is not maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since the original sequence is maximal, the only place where a new brick B could be added to the new sequence is between Bi+1 and Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have then that B ∈ Filt(Bi, Bi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since, by Hom-orthogonality, we have that HomΛ(Bi, B) = HomΛ(B, Bi+1) = 0, B cannot contain Bi as a submodule or Bi+1 as a factor module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We let B = L0 ⊃ L1 ⊃ · · · ⊃ Ls = 0 be the composition series of B in terms of Bi and Bi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We let Fi = Li−1/Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then we must have that F1 ∼= Bi and Fs ∼= Bi+1 by backwards Hom- orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, there must be some j such that Fj ∼= Bi and Fj+1 ∼= Bi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We can swap the order of these two factors in the composition series if the subquotient Lj−1/Lj+1 ∼= Bi ⊕ Bi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If we continue making such swaps where possible, we must eventually reach a case where Lj−1/Lj+1 is not isomorphic to such a direct sum, since we cannot have a composition series with F1 ∼= Bi+1 or Fs ∼= Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Thus, there exists a module B′ = Lj−1/Lj+1 with a non-split short exact sequence 0 → Bi+1 → B′ → Bi → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This gives that Ext1 Λ(Bi, Bi+1) ̸= 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ The following observation will be used in several proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that G and G′ are distinct maximal green sequences such that B(G) ⊇ B(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then there exists a pair of bricks B, B′ which are adjacent with B < B′ in G′, but which have B′ < B in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' There must be a pair of bricks of G′ which are ordered differently under G, since we are assuming that G and G′ are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Thus, choose a pair of A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 25 bricks B and B′ of G′ which are ordered differently in G, such that B and B′ are as close as possible in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If there is a brick B′′ of G′ between B and B′, then one of the pairs (B, B′′) and (B′′, B′) must be ordered differently in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But this contradicts the choice of B and B′ as the closest bricks which are ordered differently between the two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Characterising equivalent maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Our prelimi- nary work now puts us in a position to give different characterisations of equiv- alent maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The following theorem gives us six equivalent criteria for when a pair of maximal green sequences are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be a finite-dimensional algebra over a field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G1 and G2 be two maximal green sequences of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) G1 and G2 are related by a finite sequence of deformations across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) E(G1) = E(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) Ss(G1) = Ss(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (4) Sτ(G1) = Sτ(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (5) For any Λ-module M, SSFG1(M) = SSFG2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (6) For any Λ-module M, SFG1(M) = SFG2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The fact that these six notions of equivalence of maximal green sequences are the same indicates that this is the “correct” equivalence relation to impose upon maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' See also Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='23 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='24 for further explanation of the intuition behind this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given a finite-dimensional algebra Λ over a field K with G and G′ two maximal green sequences of Λ, we say that G and G′ are equivalent if any one of the six interchangeable conditions from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If G and G′ are equivalent, then we write G ∼ G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We first show that, under one of the conditions in the statement of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20, maximal green sequences have the same bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If G and G′ are maximal green sequences such that for any Λ- module M we have SFG(M) = SFG′(M), then we have that B(G) = B(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let B ∈ B(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then {B} = SFG(B) = SFG′(B), by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We must then have B ∈ B(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ We now prove our first main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will prove the implications (1) ⇒ (2) ⇒ (3) ⇒ (1), (1) ⇒ (5) ⇒ (6) ⇒ (1), 26 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS and the equivalence (3) ⇔ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is immediate from [AIR14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2] that (3) and (4) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' That (2) implies (3) is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is also immediate that (1) implies (2) since deforming across a square does not change the set of exchange pairs by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Finally, it is clear that (5) implies (6), since the set of semistable factors determines the set of stable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now show that (3) implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that we have maximal green sequences G1, G2 such that Ss(G1) = Ss(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We will prove that G1 and G2 can be deformed into each other across squares by induction on the distance between the first point where they diverge and the end of the maximal green sequences at Λ[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The base case is when this distance is zero, so that G1 = G2, in which case it is trivial that the two maximal green sequences are related by deformations across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Now suppose that G1 ̸= G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consider the first exchange pairs where G1 and G2 diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let this exchange pair be (X1, Y1) for G1 and (X2, Y2) for G2 and let T be the last two-term silting complex they share before they diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5, we must actually also have (X1, Y1) ∈ E(G2) and (X2, Y2) ∈ E(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, we know from the fact that (X1, Y1) is an exchange pair for G1 that X1 is the unique indecomposable object (up to isomorphism) in Ss(G1) ∩ T ⊥1 = Ss(G2)∩T ⊥1 such that HomKb(proj Λ)(Y1, X1[1]) ̸= 0, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5(2) on G2, we obtain that (X1, Y1) ∈ E(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The mirror-image of this argument shows that (X2, Y2) ∈ E(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We claim that, by deforming across squares, we can make (X2, Y2) the ex- change pair before (X1, Y1) in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that we cannot deform (X2, Y2) back past some exchange pair (A, B) in G1 which occurs after T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='14, we have that HomKb(proj Λ)(Y2, A[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then know that we cannot have A as a summand of T, since this would mean we cannot ex- change X2 for Y2 in T as part of G2, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We further know that A ∈ Ss(G1) ∩ T ⊥1 by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2, since A occurs in G1 after T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5, we know that X2 is the unique indecomposable in Ss(G2) ∩ T ⊥1 = Ss(G1) ∩ T ⊥1 such that HomKb(proj Λ)(Y2, X2[1]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But then A ∼= X2, which contradicts Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence we may deform G1 across squares to obtain a maximal green sequence G′ 1 where (X1, Y1) is the exchange pair immediately following (X2, Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But then, G′ 1 and G2 agree on a longer initial segment than G1 and G2, so, by the induction hypothesis, we can deform G′ 1 into G2 across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This then gives that we can deform G1 into G2 across squares, which establishes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now show that (1) implies (5) by showing that the factors of Harder– Narasimhan filtrations are preserved by deformations across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We start A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 27 with a maximal backwards Hom-orthogonal sequence of bricks G1 B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi, Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br and deform across a square to obtain a sequence G2 given by B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi−1, Bi+1, Bi, Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='18, we have that HomΛ(Bi, Bi+1) = HomΛ(Bi+1, Bi) = Ext1 Λ(Bi, Bi+1) = Ext1 Λ(Bi+1, Bi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We let M be a Λ-module and show that SSFG1(M) = SSFG2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If Filt(Bi) ∩ SSFG1(M) = ∅ or Filt(Bi+1) ∩ SSFG1(M) = ∅, then the same filtration is also the Harder–Narasimhan filtration of M by G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose then that the Harder– Narasimhan filtration of M by G1 is M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0 where Fk = Mk−1/Mk ∈ Filt(Bjk) for some jk for all 1 ⩽ k ⩽ l, with j1 < j2 < · · · < jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that jk = i and jk+1 = i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since Ext1 Λ(Bi, Bi+1) = 0, the subquotient Mk−1/Mk+1 is isomorphic to Fk ⊕ Fk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, we may replace Mk by M′ k such that Mk−1/M′ k = Fk+1 and M′ k/Mk+1 = Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The filtration obtained is then the Harder–Narasimhan filtration of M by G2, since we have Fk+1 ∈ Filt(Bi+1), Fk ∈ Filt(Bi) with Fk+1 occurring before Fk in the filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This has the same factors as the Harder–Narasimhan filtration of M by G1, so that SSFG1(M) = SSFG2(M), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Finally, we show that (6) implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Note first that this implies that B(G1) = B(G2) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We now show that one can deform G1 across squares to obtain G2 by swapping adjacent bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='19, we have adjacent bricks B < B′ in G1 which are ordered B′ < B in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since both sequences are backwards Hom-orthogonal, we have HomΛ(B, B′) = HomΛ(B′, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If we have Ext1 Λ(B, B′) ̸= 0, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='16, we have that there is a brick M which is given by a non-split extension 0 → B′ → M → B → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But then, by uniqueness of Harder–Narasimhan filtrations, this short exact sequence must give the Harder–Narasimhan filtration of M with respect to G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' However, the Harder– Narasimhan filtration of M with respect to G2 cannot have the same bricks, otherwise there is a non-split extension 0 → B → M → B′ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This gives a endomorphism M → B′ → M of M which is neither zero nor an isomorphism, contradicting the fact that M is a brick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This contradicts the assumption that 28 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS the Harder–Narasimhan filtrations with respect to G1 and G2 must have the same multisets of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, we have that Ext1 Λ(B, B′) = 0, so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='18, we can deform G1 across a square by swapping B and B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By repeating this process, we eventually deform G1 into G2 across a sequence of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Note that, if Λ is a Jacobian algebra of a quiver with potential, there is a cluster algebra associated to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Its cluster variables will be in bijec- tion with the reachable indecomposable presilting complexes in K[−1,0](proj Λ) [AIR14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' BY13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' CK06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Condition (3) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20 means that maximal green sequences are equivalent in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='21 if and only if the corresponding sets of cluster variables appearing in the clusters in the mutation sequences coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is shown in [Rei10] that, given a path algebra of a simply-laced Dynkin diagram, maximal green sequences correspond to products of quantum dilogarithms and that the value of this product is in fact independent of the maximal green sequence chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' These products of quantum dilogarithms give so-called “refined Donaldson–Thomas invariants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' See also [Kel11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' KD20] for more general statements and further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences which are related by deformation across a square correspond to products of quantum dilogarithms which differ by swapping two adjacent commuting terms in the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, maximal green sequences which are equivalent are related by finitely many such swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is natural to consider such products as the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is analogous to considering reduced words of longest elements of Weyl groups up to commutation, as we will explore in a sequel paper [GW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Thanks to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22 implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If G ∼ G′ are equivalent maximal green sequences, then we have that B(G) = B(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The converse is false, as is shown in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Maximal green sequences which have the same set of bricks are not necessarily equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, our notion of equivalence does not coincide with the notion of “weak equivalence” from [Qiu15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consider the path algebra of the quiver 1 ← 2 → 3, A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 29 where we use the convention of composing arrows as if they were functions, so that α−→ β−→= βα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The Auslander–Reiten quiver of this algebra is 2 2 3 2 1 2 1 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 3 The lattice of its two-term silting complexes is shown in Figure 2 and the lattice of its torsion classes is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consider the two maximal green sequences given by the maximal backward Hom-orthogonal sequences of bricks 2, 2 1, 1, 2 3, 3 and 2, 2 3, 3, 2 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' These two maximal green sequences have the same set of bricks, but inspection shows that one cannot transform one into the other by deforming across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, these two maximal green sequences have different sets of summands, namely � 2 1, 2, 2 3, 2 1 3, 3, 2 1[1], 2[1], 2 3[1] � and � 2 1, 2, 2 3, 2 1 3, 1, 2 1[1], 2[1], 2 3[1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22, one can regard having the same stable factors as an augmentation of the condition of having the same bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This augmentation is required to determine the equivalence class of the maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In some ways it is surprising that maximal green sequences may have the same set of bricks whilst having different sets of τ-rigids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1 shows that the τ-rigids form the relative projectives of the torsion classes, whilst Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='7 shows that the bricks form the relative simples of the torsion classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence there is no duality between simples and projectives on the level of maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is known from [Eno22, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='17] that, in general, there is no duality between simples and projectives within torsion classes — see also [Eno22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' But it is still not obvious why the bricks of a maximal green sequence should give less information than the τ-rigids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' As shown in [Eno22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='15], the existence of the duality between simples and projectives — at least in the sense of having the same number of simples as indecomposable projectives — in a functorially finite torsion class T 30 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The lattice of two-term silting complexes from Exam- ple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='26 2 1 ⊕ 2 ⊕ 2 3 2 1[1] ⊕ 2 ⊕ 2 3 2 1 ⊕ 2 1 3 ⊕ 2 3 2 1 ⊕ 2 ⊕ 2 3[1] 3 ⊕ 2 1 3 ⊕ 2 3 2 1 ⊕ 2 1 3 ⊕ 1 2 1[1] ⊕ 3 ⊕ 2 3 3 ⊕ 2 1 3 ⊕ 1 2 3[1] ⊕ 2 1 ⊕ 1 3 ⊕ 2[1] ⊕ 1 2 1[1] ⊕ 3 ⊕ 2[1] 2 1[1] ⊕ 2 ⊕ 2 3[1] 2 3[1] ⊕ 2[1] ⊕ 1 2 1[1] ⊕ 2[1] ⊕ 2 3[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' in mod Λ, for an Artin algebra Λ, is equivalent to the Jordan–H¨older property (JHP) for this torsion class considered as an exact category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Here the exact structure is the one induced by the embedding T ֒→ mod Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Enomoto shows that functorially finite torsion classes of Nakayama algebras possess the JHP [Eno22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' we show in Section 5 that for these algebras the set of bricks does determine the equivalence class of the maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One may wonder whether the set of bricks determines the equivalence class of the maximal green sequence if and only if every functorially finite torsion class in mod Λ satisfies the JHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' [Eno21, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='17] shows that the “only if” part cannot be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Namely, the example in [Eno21] gives a torsion class in the preprojective algebra of type D4 which does not satisfy the JHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' On the other hand, in a sequel to this paper [GW], we will show that the converse of A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 31 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The lattice of torsion classes from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='26 with brick labelling add � 2 1, 2 1 3, 2 3, 1, 2, 3 � add � 2, 2 3, 3 � add � 2 1, 2 1 3, 2 3, 1, 3 � add � 2, 2 1, 1 � add � 2 1 3, 2 3, 1, 3 � add � 2 1, 2 1 3, 1, 3 � add � 2 3, 3 � add � 2 1 3, 1, 3 � add � 2 1, 1 � add � 1, 3 � add � 3 � add � 2 � add � 1 � {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 1 2 3 2 1 2 3 2 2 3 1 2 3 1 2 1 3 2 1 3 2 3 2 1 1 3 3 2 1 32 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22 holds for all preprojective algebras of Dynkin type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence, there are examples where the converse of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='22 holds whilst there exist functorially finite torsion classes for which the JHP fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We conjecture that the “if” part is true, motivated by the case of Nakayama algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be an Artin algebra such that every functorially finite torsion class in mod Λ satisfies the JHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then two maximal green sequences G and G′ have the same set of bricks if and only they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Note that the JHP does not hold for some functorially finite torsion classes of the non-linearly oriented A3 algebra considered in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, in the torsion class add � 2 1, 2 1 3, 2 3, 1, 3 � the brick 2 1 3 has two different composition series: one with factors 3 and 2 1, and the other with factors 1 and 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consequently, there is no bijection between relative projectives and relative sim- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' There are three relative projectives 2 1, 2 1 3, and 2 3, while there are four relative simples 2 1, 2 3, 1, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Another plausible definition of equivalence of maximal green se- quences which fails to coincide with those of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='20 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5 that a maximal green sequence can be specified as the sequence of vertices mutated at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One might conjecture that two maximal green sequences are equivalent if and only if the underlying multisets of these sequences coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is clear that these multisets are preserved by deformation across squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' How- ever, these multisets may coincide for inequivalent maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For instance, the maximal green sequence down the left-hand side of Figure 2 has sequence 1, 2, 3, 2, while the sequence down the right-hand side has sequence 3, 2, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 33 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' An increasing elementary polygonal deformation, where X and A are indecomposable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' E ⊕ X ⊕ A E ⊕ Z ⊕ A E ⊕ Z ⊕ C E ⊕ X ⊕ C′ E ⊕ X′ ⊕ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The underlying multisets are the same here, but the maximal green sequences are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partial orders on equivalence classes The equivalence relation on maximal green sequences defined in the previous section reveals more structure on the set of maximal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, this equivalence relation allows one to define partial orders on the equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In this section, we define three such orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The first uses deformations across oriented polygons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' the second uses reverse-inclusion of summands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' the third uses refinement of Harder–Narasimhan filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We show that the first order implies the second and the third, but we conjecture that the three actually coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across oriented polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The first of these partial or- ders has covering relations given by deformations across oriented polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let G and G′ be maximal green sequences of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If G and G′ only differ in that G contains the path of length greater than two around an oriented polygon, whilst G′ contains the length-two path, then we say that G′ is an increasing elementary polygonal deformation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' By extension, we also say that [G′] is an increasing elementary polygonal deformation of [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Similarly, we say that G is a decreasing elementary polygonal deformation of G′ and that [G] is a decreasing elementary polygonal deformation of [G′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Note that an increasing elementary polygonal deformation decreases the length of the maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One can think of it instead as increasing the speed of the maximal green sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Deformations across oriented polygons in terms of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Just as we in- terpreted deformations across squares in terms of maximal backwards Hom- orthogonal sequences of bricks, we also wish to do the same for increasing ele- mentary polygonal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 34 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Let Λ be a finite-dimensional algebra over a field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Suppose that V T U V′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' V′ r M L B1 B2 Br−1 Br is an oriented polygon in ff-tors Λ with its brick labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then B1 ∼= M and Br ∼= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have that [T , U] = Filt(L, M) = Filt(B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br), recalling the notation [T , U] from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We then have that L and M must be precisely the relatively simple objects of [T , U], since neither brick can admit a filtration with the other as factors without violating the brick condition or the backwards Hom-orthogonality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It then follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6 that L and M must occur as elements of the set {B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' It is clear that L cannot occur before M amongst these bricks, otherwise L, M cannot be a maximal backwards Hom-orthogonal sequence of bricks in [T , U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If we do not have B1 ∼= M and Br ∼= L, then we also get a contradiction to backwards Hom-orthogonality, since all of B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br have filtrations with L and M as factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ One can view an increasing elementary polygonal deformation as swapping maximal green sequences in an abelian category with two simple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Such a category has at most two maximal green sequences up to equivalence, and in our case it has precisely two, where one has length two and the other is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In the situation of the polygon from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) Filt(L, M) is an abelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) The two paths around the polygon give two maximal green sequences of Filt(L, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (3) The two paths around the polygon give two different sets of Harder– Narasimhan filtrations of Filt(L, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We have HomΛ(L, M) = HomΛ(M, L) = 0 by backwards Hom-orthogon- ality of maximal green sequences going through the sides of the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then (1) follows from [Rin76, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2], which states that Filt(−) of a set of Hom-orthogonal bricks is an abelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For (2), it follows from the definition of brick labels that the sequences of bricks given by the two paths around the polygon must be backwards Hom-orthogonal in Filt(L, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For (3), the fact that the two paths around the polygon give Harder–Narasimhan filtrations on Filt(L, M) A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 35 then follows from (2) by [Igu20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Tre20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The fact that the two sets of Harder– Narasimhan filtrations must be different then follows from the fact that in the longer path around the polygon, B2 is the only factor in its filtration, whilst this cannot be the case for the shorter path around the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ The interpretation of increasing elementary polygonal deformations is then as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A maximal green sequence G′ is an increasing elementary polygonal deformation of a maximal green sequence G if and only if, as maximal backwards Hom-orthogonal sequences of bricks, we have that G′ is B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br whilst G is B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Bi−1, Bi+1, B′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , B′ s, Bi, Bi+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br for s ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The forwards direction is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' For the backwards direction, let T be the basic two-term silting complex corresponding to the torsion class Tors(Bi, Bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br) and T ′ be the basic two-term silting complex correspond- ing to the torsion class Tors(Bi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' , Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then T ′ is obtained from T by two green mutations since the corresponding torsion classes differ by two minimal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence T ∼= E ⊕ X ⊕ X′ and T ′ ∼= E ⊕ Y ⊕ Y ′ where X, X′, Y , and Y ′ are all indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9, the only other path from T to T ′ in 2-silt Λ is the other path around the polygon determined by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This must be the path taken in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since s ⩾ 1, the polygon determined by E is ori- ented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Hence G and G′ only differ in that G′ contains the length two path around the oriented polygon whilst G contains the longer path, so G′ is an increasing elementary polygonal deformation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Partial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We can now define the three partial orders on equivalence classes of maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (1) The partial order ⩽� on equivalence classes of maximal green sequences is defined via its covering relations, which are that [G]⋖� [G′] if and only if [G′] is an increasing elementary polygonal deformation of [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We refer to this as the deformation partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' (2) The partial order ⩽S is defined via [G] ⩽S [G′] if and only if Ss(G) ⊇ Ss(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is evidently also equivalent to having Sτ(G) ⊇ Sτ(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We refer to this as the summand partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 36 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS (3) The partial order ⩽HN is defined by [G] ⩽HN [G′] if and only if, for any module M, we have SFG′(M) = � B∈SFG(M) SFG′(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If [G] ⩽HN [G′], then we say that the G′-HN filtrations refine the G-HN filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Informally, [G] ⩽HN [G′] if the G′-stable factors of any Λ- module M can be obtained by breaking up the G-stable factors of M into their G′-stable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We refer to this as the Harder–Narasimhan or HN partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' As shown in [Wil22a, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4] [Wil22b, The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='6], the deformation order here should be seen as a higher- dimensional incarnation of the order on silting complexes given by green muta- tion, whilst the summand partial order should be seen as a higher-dimensional incarnation of the order on silting complexes given by inclusion of aisles [AI12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' These orders are known to have the same Hasse diagram [AI12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Analogous orders exist on tilting modules [RS91, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='2] and support τ-tilting pairs [AIR14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='4], which are likewise known to have the same Hasse diagram [HU05, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='1] [AIR14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The Harder–Narasimhan order is new and does not have an analogue on silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The other new feature here is of course that one must introduce the equivalence relation on maximal green sequences in order to see the partial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Note that a partial order on equivalence classes is exactly the same thing as a preorder, so one could instead consider preorders on maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' We prefer to keep the equivalence relation and the partial orders conceptually separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The reason why we must use the stable factors rather than the semistable factors to define the Harder–Narasimhan order is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The plausible alternative definition using the semistable factors would be that [G] ⩽HN [G′] if and only if for all Λ-modules M, we have that SSFG′(M) = � F ∈SSFG(M) SSFG′(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consider the path algebra of the A2 quiver 1 ← 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This has two maximal green sequences, namely G given by the sequence of bricks 2, 2 1, 1, and G′ given by the sequence of bricks 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES 37 Then we have that [G] ⩽HN [G′], but we have SSFG′ � 2 ⊕ 2 1� = {1, 2 ⊕ 2} ̸= {1, 2, 2} = SSFG′ � 2 1� ⊔ SSFG′(2) = � F ∈SSFG � 2⊕2 1� SSFG′(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In order for this to work correctly, we need to break up 2 ⊕ 2 into 2, 2 by con- sidering stable factors rather than semistable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Given two maximal green sequences G and G′ such that [G] ⩽HN [G′], one might wonder whether, up to equivalence, the G′-HN filtration of a Λ-module M can be obtained from the G-HN filtration of M by breaking up the G-semistable factors according to their G′-HN filtrations, without doing any rearranging of the orders of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' To be more precise, suppose that, up to equivalence, the G-HN filtration of M is M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0 with Fi := Mi−1/Mi, and suppose that the G′-HN filtration of each Fi is Fi = Li0 ⊃ Li1 ⊃ · · · ⊃ Lili = 0 with Hij := Li(j−1)/Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' One might hope that, again up to equivalence, the G′-HN filtration of M is M = M10 ⊃ M11 ⊃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' M1(l1−1) ⊃ M20 ⊃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' ⊃ Mi0 ⊃ · · · ⊃ Mi(li−1) ⊃ · · · ⊃ Ml(ll−1) = 0 where Mi(j−1)/Mij = Hij for 1 ⩽ i ⩽ l and 0 ⩽ j ⩽ li −2 and Mi(li−1)/M(i+1)0 = Hili for 1 ⩽ i ⩽ l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Unfortunately, this is not generally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' In general, one has to reorganise the semistable factors Hij to obtain the G′-HN filtration, even if one replaces G and G′ by equivalent maximal green sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' This is shown in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Consider the path algebra of the following algebra of type �A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' 2 4 1 3 This algebra has a maximal green sequence G given by the sequence of bricks 1, 1 2, 1 4, 1 2 4, 3, 3 2, 2, 3 4, 4 38 MIKHAIL GORSKY AND NICHOLAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' WILLIAMS and a maximal green sequence G′ given by the sequence of bricks 4, 2, 3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Since the bricks of G′ are precisely the simple modules, it is clear that we have [G] ⩽HN [G′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Now consider the module M = 1 3 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The G-HN filtration of M is 1 3 2 4 ⊃ 3 4 ⊃ 0, with SSFG(M) = � 1 2, 3 4� , whilst the G′-HN filtration of M is 1 3 2 4 ⊃ 1 3 2 ⊃ 1 ⊕ 3 ⊃ 1 ⊃ 0, with SSFG′(M) = {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Now, if we were to try to construct the G′-HN filtration of M as above, by sticking together the G′-HN filtration of 1 2 with the G′-HN filtration of 3 4 , then the G′-semistable factors would appear in the order 2, 1, 4, 3, whereas in actuality they appear in the order 4, 2, 3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Moreover, no amount of deformation across squares for either G or G′ can change this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Indeed, the G-HN filtration of M is unique in the equivalence class, whilst deformation across squares cannot change the order in which 1 and 4 occur in G′, since Ext1 Λ(4, 1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' The HN order on maximal green sequences implies inclusion of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' If [G] ⩽HN [G′], then B(G) ⊇ B(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf'} +page_content=' Furthermore, if [G]