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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf,len=1985
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='01498v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='RT]  4 Jan 2023  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' g-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined  by tilting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We give a classification of complete g-fans of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  More explicitly, our  first main result asserts that every complete sign-coherent fan of rank 2 is a g-fan of some  finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Our proof is based on three fundamental results, Gluing Theorem,  Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level  of finite dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Our second main result gives a necessary and sufficient condition  for algebras of rank 2 to be g-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Preliminaries  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Preliminaries on fans  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Sign-coherent fans of rank 2  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Basic results in silting theory  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Preliminaries  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Silting complexes in terms of matrices  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Uniserial property of g-finite algebras  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Gluing, Rotation and Subdivision of g-fans  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Gluing fans  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Rotation and Mutation  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Subdivision and Extension  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3  22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Gluing fans II  23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  g-Convex algebras of rank 2  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Characterizations of g-convex algebras of rank 2  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1  27  Acknowledgments  29  References  29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Introduction  The notion of tilting complexes is central to control equivalences of derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The  class of silting complexes [KV] gives a completion of the class of tilting complexes with respect to  mutation, which is an operation to replace a direct summand of a given silting complex to construct  a new silting complex [AI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The subclass of 2-term silting complexes enjoys remarkable properties  [AIR, DF].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' They give rise to a fan in the real Grothendieck group of a finite dimensional algebra  A, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [H1, H2, Pl, B, DIJ, BST, As].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In our previous article [AHIKM1], we introduced a g-fan Σ(A) of A and established a basic  theory of g-fans and the associated g-polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A g-fan of each finite dimensional algebra A  belongs to the following special class of nonsingular fans [AHIKM1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A sign-coherent fan is a pair (Σ, σ+) satisfying the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  1  2  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  (a) Σ is a nonsingular fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) σ+, −σ+ ∈ Σd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) Take a Z-basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ed of Zd such that σ+ = cone{ei | 1 ≤ i ≤ d}, and denote the orthant  corresponding to ǫ ∈ {±1}d by  Rd  ǫ := cone{ǫ(1)e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ǫ(d)ed} = {x1e1 + · · · + xded | ǫ(i)xi ≥ 0 for each 1 ≤ i ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then for each σ ∈ Σ, there exists ǫ ∈ {±1}d such that σ ⊆ Rd  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote by Fansc(d) the set of complete sign-coherent fans in Rd, and by k-Fan(d) the set of  complete g-fans of finite dimensional k-algebras of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Note that a g-fan Σ(A) is complete if  and only if A is g-finite (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have  Fansc(d) ⊃ k-Fan(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  It is very natural to study the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Characterize complete sign-coherent fans in Rd which can be realized as a g-fan of  some finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  This paper is devoted to give a complete answer to this problem for the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The result  was very simple and came as a surprise to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For each field k, we have  Fansc(2) = k-Fan(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus any complete sign-coherent fan in R2 can be realized as a g-fan of some finite dimensional  k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We explain our method to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Each sign-coherent fan of rank 2 is obtained by  gluing two fans of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ =  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  Σ′ =  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  Recall that a finite dimensional k-algbera Λ is elementary if the k-algebra Λ/JΛ is isomorphic to  a product of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This is automatic if Λ is basic and k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We prove Gluing  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1, which asserts that if both Σ and Σ′ are g-fans of finite dimensional elementary k-  algebras, then so is their gluing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Therefore by symmetry, it suffices to consider sign-coherent fans  Σ of the form above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Now such Σ can be obtained from the fan  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  by applying subdivision in the fourth quadrant repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We prove Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3 and  Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7, which imply that if Σ is a g-fan of a finite dimensional k-algebra, then  so are the subdivisions of Σ in the fourth quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Figure 1 gives fans in Fan+−  sc (2) with at most 8 facet, where each edge shows a subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Figure 2 gives examples of algebras whose g-fans are given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For each finite dimensional algebra A, we define a g-polytope P(A) by gluing each simplex  associated with the cones in Σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  If P(A) is convex, we call Σ(A) convex and A g-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For example, Brauer tree algebras A are g-convex, and this fact plays an important role in the  classification of 2-term tilting complexes of A [AMN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' From tilting theoretic point of view, g-convex  algebras are the most fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Therefore it is important to study the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Classify convex g-fans in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  3  Σ00  Σ111  Σ2121  Σ1212  Σ31221  Σ22131  Σ12213  Σ21312  Σ13122  Σ412221  Σ321321  Σ313131  Σ312312  Σ231231  Σ222141  Σ221412  Σ122214  Σ123123  Σ214122  Σ131313  Σ213213  Σ132132  Σ141222  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Fans in Fan+−  sc (2) with at most 8 facets  An answer to the case d = 2 was given in [AHIKM1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' There are precisely 7 convex  g-fans up to isomorphism of g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  +  −  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❖❖❖❖❖❖❖  ❄❄❄❄❄  +  −  ❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄  ❖❖❖❖❖❖❖  ❄❄❄❄❄  +  −  ❄❄❄❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖❖❖❖❖❖❖  ❄���❄❄❄  +  −  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄  ✴✴✴✴✴✴✴  ❄❄❄❄❄  ❖❖❖❖❖❖❖  ❄❄❄❄❄  More precisely, in the last Section 5, we show that there are 16 convex g-fans in Fansc(2)  Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b with a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We also give a characterization of algebras whose g-fans are one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' ab1 − c1a  �  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Algebras whose g-fans are given in Figure 1  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  5  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a basic finite dimensional algebra, {e1, e2} a complete set  of primitive orthogonal idempotents in A, and Pi = eiA (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some  left generator x of e1Ae2 and Rx := {a ∈ e1Ae1 | ax ∈ xAe2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some  right generator x of e1Ae2 and Lx := {b ∈ e2Ae2 | xb ∈ e1Ax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ⑧⑧⑧⑧⑧⑧  Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄  ❄  ❄  ❄  ❄  ❄  ✴✴✴✴✴✴✴✴✴  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  Further, in a forthcoming paper [AHIKM2], we will give a complete answer to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4 for  d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Preliminaries on fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We recall some fundamental materials on fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We refer the reader  to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [F, BR, BP] for these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  A convex polyhedral cone σ is a set of the form σ = {���s  i=1 rivi | ri ≥ 0}, where v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vs ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote it by σ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Note that {0} is regarded as a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We  collect some notions concerning convex polyhedral cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let σ be a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The dimension of σ is the dimension of the linear space generated by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We say that σ is strongly convex if σ ∩ (−σ) = {0} holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', σ does not contain a linear  subspace of positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call σ rational if each vi can be taken from Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote by ⟨·, ·⟩ the usual inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  A supporting hyperplane of σ is a hyperplane  {v ∈ σ | ⟨u, v⟩ = 0} in Rd given by some u ∈ Rd satisfying σ ⊂ {v ∈ Rd | ⟨u, v⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  A face τ of σ is the intersection of σ with a supporting hyperplane of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In what follows, a cone means a strongly convex rational polyhedral cone for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A fan Σ in Rd is a collection of cones in Rd such that  (a) each face of a cone in Σ is also contained in Σ, and  (b) the intersection of two cones in Σ is a face of each of those two cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For each i ≥ 0, we denote by Σi the subset of cones of dimension i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For example, Σ0 consists of  the trivial cone {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We call each element in Σ1 a ray of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We collect some notions concerning fans used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ be a fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call Σ finite if it consists of a finite number of cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call Σ complete if �  σ∈Σ σ = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call Σ nonsingular (or smooth) if each maximal cone in Σ is generated by a Z-basis for Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We prepare some notions which will be used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ be a nonsingular fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We call Σ pairwise positive if the following  condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  6  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  For each two adjacent maximal cones σ, τ ∈ Σd, take Z-basis {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vd−1, vd} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vd−1, v′  d}  of Zd such that σ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vd−1, vd} and τ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vd−1, v′  d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then vd + v′  d belongs  to cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vd−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ and Σ′ be fans in Rd and Rd′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (1) An isomorphism Σ ≃ Σ′ of fans is an isomorphism Zd ≃ Zd′ of abelian groups such that  the induced linear isomorphism Rd → Rd′ gives a bijection Σ ≃ Σ′ between cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2) Let (Σ, σ+) and (Σ′, σ′  +) be sign-coherent fans in Rd and Rd′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' An isomorphism  of sign-coherent fans is an isomorphism f : Σ ≃ Σ′ of fans such that {f(σ+), f(−σ+)} =  {σ′  +, −σ′  +}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Sign-coherent fans of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we introduce some terminologies of sign-  coherent fans of rank 2, and discuss some fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Let Σ be a complete nonsingular fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We denote the rays of Σ by  v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vn−1, vn = v0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  which are indexed in a clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For each 1 ≤ i ≤ n, since Σ is nonsingular, there  exists an integer ai satisfying  aivi = vi−1 + vi+1 for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call the sequence of integers  s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2)  the defining sequence of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In fact, Σ is uniquely determined by its defining sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A fan with  defining sequence (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an) is denoted by  Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [F, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5] An integer sequence (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an) is a defining sequence of nonsingular  complete fan of rank 2 if and only if it satisfies  n  �  i=1  ai = 3n − 12 and  �0  −1  1  a1  � �0  −1  1  a2  �  · ·  �0  −1  1  an  �  =  �1  0  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We denote by Fansc(2) the set of all (possibly infinite) fans Σ satisfying that  Σ is a sign-coherent fans (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1) of rank 2 with positive and negative cones  σ+ := cone{(1, 0), (0, 1)} and σ− := cone{(−1, 0), (0, −1)} respectively,  each ray is a face of precisely two facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote the subset of complete fans by  Fansc(2) ⊂ Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For Σ ∈ Fansc(2), we denote the rays of Σ in a clockwise orientation by  Σ1 = {v1 := (1, 0), v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vn−1, vn = v0 := (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then there exists 2 ≤ i ≤ n − 2 such that vi = (0, −1) and vi+1 = (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' vn=v0=(0,1)  v1=(1,0)  vi=(0,−1)  vi+1=(−1,0)  +  −  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  In this case, it is more convenient to rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2) as  s(Σ) = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' an, an−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus we mainly use the notation  Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai+1) = Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=',ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='an,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=',ai+1  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  7  instead of Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We consider subsets  Fan  +−  sc (2)  ⊂  Fansc(2)  ∪  ∪  Fan+−  sc (2)  ⊂  Fansc(2)  which consist of fans Σ containing σ−+ := cone{(−1, 0), (0, 1)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Σ has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' σ−+  +  −  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Thus the rays and the facets of Σ ∈ Fan+−  sc (2) are written as  Σ1  =  {v1 = (1, 0), v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vn−2 = (0, −1), vn−1 = (−1, 0), vn = v0 = (0, 1)},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3)  Σ2  =  {σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , σn−3, σn−2 = σ−, σn−1 = σ−+, σn = σ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4)  Similarly, we define Fan  −+  sc (2) and Fan−+  sc (2) as the subsets of Fansc(2) and Fansc(2) respectively  which consist of fans containing σ+− := cone{(1, 0), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following observations are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (1) The correspondence Σ �→ {−σ | σ ∈ Σ} gives bijections Fan  +−  sc (2) → Fan  −+  sc (2) and Fan+−  sc (2) →  Fan−+  sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2) Let Σ ∈ Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Σ ∈ Fan+−  sc (2) (respectively, Σ ∈ Fan−+  sc (2)) holds if and only if s(Σ)  has the form  (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) (respectively, (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , bm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In this case, bi ≥ 0 holds for any 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For Σ ∈ Fan  +−  sc (2) and Σ′ ∈ Fan  −+  sc (2), we define Σ ∗ Σ′ ∈ Fansc(2) by  (Σ ∪ Σ′)1  :=  Σ1 ∪ Σ′  1  (Σ ∪ Σ′)2  :=  (Σ2 \\ {σ−+}) ∪ (Σ′  2 \\ {σ+−}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ = +  −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  σ−+  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Σ′ = +  −  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  σ+−  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Σ ∗ Σ′ = +  −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  Then, we clearly have  Fansc(2)  =  Fan  +−  sc (2) ∗ Fan  −+  sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan  +−  sc (2), Σ′ ∈ Fan  −+  sc (2)},  Fansc(2)  =  Fan+−  sc (2) ∗ Fan−+  sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan+−  sc (2), Σ′ ∈ Fan−+  sc (2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5)  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ be a (possibly infinite) nonsingular fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For a cone σ := cone{u, v}  of Σ, we define a new nonsingular fan Dσ(Σ) by  Dσ(Σ)1  =  Σ1 ∪ {cone{u + v}},  Dσ(Σ)2  =  (Σ2 \\ {σ}) ⊔ {cone{u, u + v}, cone{v, u + v}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call Dσ(Σ) the subdivision of Σ at σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='.  σ  ❨  ❨  ❨  ❨  ❨  ❨  ❨  ❡  ❡  ❡  ❡  ❡  ❡  ❡  Dσ(Σ) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='. ❨  ❨  ❨  ❨  ❨  ❨  ❨  ❡  ❡  ❡  ❡  ❡  ❡  ❡  ❡❡❡❡❡❡❡  ❨❨❨❨❨❨❨  8  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  For a sequence a = (a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an) and 1 ≤ i ≤ n, we define a new sequence by  Di(a) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai−1, ai + 1, 1, ai+1 + 1, ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6)  For a complete nonsingular fan Σ with rays (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1) and σi := cone{vi, vi+1} for 1 ≤ i ≤ n, we have  s ◦ Dσi(Σ) = Di ◦ s(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7)  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Figure 1 gives fans in Fan+−  sc (2) with at most 8 facets, where  Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=',an := Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0)  and each edge shows a subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Figure 2 gives examples of algebras whose g-fans are given  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For example, Σ111 is the g-vector fan of a cluster algebra of type A2 [FZ1, FZ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Similarly, Σ1212 and Σ2121 are the g-vector fans of cluster algebras of type B2, and Σ131313 and  Σ313131 are the g-vector fans of cluster algebras of type G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Later we need the following observation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [F, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Each fan in Fan+−  sc (2) can be obtained from Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) by a sequence of  subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) =  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  To prove this, we need the following preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [F, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ ∈ Fan+−  sc (2) and s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If n ≥ 5, then  there exists 2 ≤ i ≤ n − 3 satisfying ai = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let vi = (xi, yi) ∈ Z2 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume that n ≥ 5 and ai ≥ 2 for any 2 ≤ i ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We claim that xi+1 ≥ xi holds for each 1 ≤ i ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In fact, n ≥ 5 implies x2 ≥ 1 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then  we have  xi+1 = aixi − xi−1 ≥ 2xi − xi−1 ≥ xi  for each 2 ≤ i ≤ n − 3, and the claim follows inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Consequently 1 = x1 ≤ x2 ≤ · · · ≤  xn−2 = 0 holds, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We are ready to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let F ⊂ Fan+−  sc (2) be the set of fans obtained from Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) by a  sequence of subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' It suffices to show Fan+−  sc (2) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We will show that each Σ ∈ Fan+−  sc (2) belongs to F by using induction on n = #Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Clearly n ≥ 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If n = 4, then Σ = Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Suppose that Σ with #Σ2 = n ≥ 5 belongs to Fan+−  sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In terms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4), there  exists 2 ≤ i ≤ n − 3 satisfying vi = vi−1 + vi+1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since vi−1, vi+1 forms a Z-basis  of Z2, we obtain a new fan Σ′ ∈ Fan+−  sc (2) by  Σ′  1  :=  Σ1 \\ {vi},  Σ′  2  :=  (Σ2 \\ {σi−1, σi}) ∪ {σ} for σ := cone{vi−1, vi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since #Σ′  2 = n − 1, the induction hypothesis implies Σ′ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus Σ = Dσ(Σ′) ∈ F holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For each n ≥ 1, we have a bijection  {Σ ∈ Fan+−  sc (2) | #Σ2 = n + 3} ≃ {the ways to parenthesize n factors completely},  where parentheses show how cones in the fourth quadrant are obtained by iterated subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For example, Σ141222 in Figure 1 has 5 cones σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , σ5 in the fourth quadrant in terms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4),  and they are parenthesized as σ1(((σ2σ3)σ4)σ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In particular, we have  #{Σ ∈ Fan+−  sc (2) | #Σ2 = n + 3} = 1  n  �2n − 2  n − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  9  We also have a bijection  {Σ ∈ Fan+−  sc (2) | #Σ2 = n + 3} ≃ {Triangulations of a regular (n + 1)-gon},  where Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=',an+1 corresponds to a triangulation satisfying the following condition: Let 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', n+  1 be the vertices of the regular (n + 1)-gon in a clockwise direction, and ai (1 ≤ i ≤ n + 1) the  number of triangles containing the vertex i in the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For example, Σ141222 corresponds  to the following triangulation, where 1 is the top vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We introduce piecewise linear transformation of sign coherent fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This is a general-  ization of mutation of g-vectors of cluster algebras of rank 2 [FZ2, NZ], and also a special case of  so called combinatorial mutation [ACGK, FH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For Σ ∈ Fan  +−  sc (2) with σ+ = cone{(0, 1), (1, 0)}, take σ = cone{(1, 0), (ℓ, −1)} ∈  Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Define a new sign-coherent fan Σ′ by  Σ′  1  :=  (Σ1 \\ {(0, 1)}) ∪ {(−ℓ, 1)}  Σ′  2  :=  (Σ2 \\ {σ+, σ−+}) ∪ {−σ, cone{(−ℓ, 1), (1, 0)}},  where the positive and negative cones of Σ′ are σ and −σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='0)  (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='−1)  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='−1)  (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='0)  +  −  σ  σ−+  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄❄❄❄❄❄  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ρ(Σ) ≃ Σ′ = (−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='0)  (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='−1)  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='−1)  (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='0)  +  −  σ  −σ  ❄❄❄❄❄❄  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  We define the rotation ρ(Σ) ∈ Fan  +−  sc (2) of Σ as the image of Σ′ by a linear transformation of R2  mapping (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0) �→ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 1) and (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' −1) �→ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We give basic properties of rotation, where the name “rotation” is explained by (a) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Σ ∈ Fan+−  sc (2) with facets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4) and s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) We have  s(ρ(Σ)) = (a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2, a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In particular, ρn−2(Σ) = Σ holds, and therefore ρ is an invertible operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) For each 1 ≤ i ≤ n − 3, we have  Dσi(Σ) = ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (a) Recall Σ1 = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , vn} and aivi = vi−1 + vi+1 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Moreover  ρ(Σ)1 = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , wn} where wi := vi+1 (i ̸= n − 1), wn−1 := −v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Hence we have  wi−1 + wi+1  =  vi + vi+2 = ai+1vi+1 = ai+1wi for i ̸= n − 2, n,  wn−1 + w1  =  −v2 + v2 = 0 · wn,  wn−3 + wn−1  =  vn−2 − v2 = −(vn + v2) = −a1v1 = a1vn−1 = a1wn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus s(ρ(Σ)) = (a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2, a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) By (a), we have s ◦ ρi+1(Σ) = (ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus  s ◦ Dσn−3 ◦ ρi+1(Σ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7)  = Dn−3 ◦ s ◦ ρi+1(Σ) = (ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai−1, ai + 1, 1, ai+1 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  10  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  By (a) again, we have  s ◦ ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ)  =  (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , ai−1, ai + 1, 1, ai+1 + 1, ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0)  =  Di ◦ s(Σ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7)  = s ◦ Dσi(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since a fan is uniquely determined by its defining sequence, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Basic results in silting theory  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a finite dimensional algebra over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let K0(proj A) be the  Grothendieck group of the additive category proj A, which is identified with the Grothendieck  group of the triangulated category Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We recall basic results on silting theory from  [AI, AIR, AHIKM1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' First we recall the definition of 2-term silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let T = (T i, di) ∈ Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) T is called presilting if HomKb(proj A)(T, T [ℓ]) = 0 for all positive integers ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) T is called silting if it is presilting and Kb(proj A) = thick T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) T is called 2-term if T i = 0 for all i ̸= 0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this case, the class [T ] = [T 0] − [T −1] ∈  K0(proj A) of T is called the g-vector of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (d) An element of K0(proj A) is rigid if it is a g-vector of some 2-term presilting complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote by siltA (respectively, psiltA, 2-siltA, 2-psiltA) the set of isomorphism classes of basic  silting (respectively, presilting, 2-term silting, 2-term presilting) complexes of Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Note  that a 2-term presilting complex T is silting if and only if |T | = |A| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For T, U ∈ siltA, we write T ≥ U if HomKb(proj A)(T, U[ℓ]) = 0 holds for all positive integers ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then (siltA, ≥) is a partially ordered set [AI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In this paper, the subposet (2-siltA, ≥) of (siltA, ≥) plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  It is known that  Hasse(2-siltA) is n-regular for n := |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' More precisely, let T = T1 ⊕ · · · ⊕ Tn ∈ 2-siltA with  indecomposable Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For each 1 ≤ i ≤ n, there exists precisely one T ′ ∈ 2-siltA such that T ′ =  T ′  i ⊕ (�  j̸=i Tj) for some T ′  i ̸= Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this case, we call T ′ mutation of T at Ti and write  T ′ = µTi(T ) = µi(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In this case, either T > T ′ or T ′ < T holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We denote T ′ by µ−  i (T ) (respectively, µ+  i (T )) if  T > T ′ and call it left mutation (respectively, right mutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The following result is fundamental  in silting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let T, T ′ ∈ 2-siltA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Take a decomposition T = T1⊕· · ·⊕Tn with indecomposable  Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then the following conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) T > T ′, and T and T ′ are mutation of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) There is an arrow T → T ′ in Hasse(2-siltA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) T ′ = T ′  i ⊕ (�  j̸=i Tj) and there is a triangle  Ti  f−→ Ui → T ′  i → Ti[1]  such that f is a minimal left (add �  j̸=i Tj)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (d) T ′ = T ′  i ⊕ (�  j̸=i Tj) and there is a triangle  Ti → Ui  g−→ T ′  i → Ti[1]  such that g is a minimal right (add �  j̸=i Tj)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The triangles in (c) and (d) are isomorphic, and called an exchange triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  To introduce the g-fan of a finite dimensional k-algebra A, we consider the real Grothendieck  group of A:  K0(proj A)R := K0(proj A) ⊗Z R ≃ R|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  11  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For T = T1 ⊕ · · · ⊕ Tℓ ∈ 2-psiltA with indecomposable Ti, let  C(T )  :=  {  ℓ  �  i=1  ai[Ti] | a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , aℓ ≥ 0} ⊂ K0(proj A)R,  C≤1(T )  :=  {  ℓ  �  i=1  ai[Ti] | a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , aℓ ≥ 0,  ℓ  �  i=1  ai ≤ 1} ⊂ K0(proj A)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We call the set  Σ(A) := {C(T ) | T ∈ 2-psiltA}  of cones the g-fan of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We also define the g-polytope P(A) of A by  P(A) :=  �  T ∈2-siltA  C≤1(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We say that A is g-convex if the g-polytope P(A) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Notice that Σ(A) can be an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We give the following basic properties of g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a finite dimensional algebra over a field k and n := |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) Σ is a pairwise positive sign-coherent fan whose positive (respectively, negative) cone is given  by σ+ := C(A) (respectively, σ− := C(A[1])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Any cone in Σ(A) is a face of a cone of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) Any cone in Σ(A) of dimension n − 1 is a face of precisely two cones of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following basic observation will be used frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be a finite dimensional algebra with orthogonal primitive idempotents  1 = e1 + e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Under the identification P1 = (1, 0) and P2 = (0, 1), the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) cone{(−1, 0), (0, 1)} ∈ Σ(Λ) if and only if e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) cone{(1, 0), (0, −1)} ∈ Σ(Λ) if and only if e1Λe2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  e2Λe1 = 0 ⇔ +  −  P1  P2  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  e2Λe1 = 0 ⇔ +  −  P1  P2  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We only prove (a): Σ(A) ∈ Fan+−  sc (2) if and only if P1[1] ⊕ P2 ∈ 2-siltA if and only if  HomKb(proj A)(P1, P2) = 0 if and only if e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We end this subsection with recalling the sign decomposition technique studied in [Ao, AHIKM1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We have to introduce the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a basic finite dimensional algebra over a field k with |A| = n, and  1 = e1 + · · · + en the orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For ǫ ∈ {±1}n, we define  K0(proj A)ǫ,R := cone(ǫi[eiA] | i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', n})  and a subfan of Σ(A) by  Σǫ(A) := {σ ∈ Σ(A) | σ ⊂ K0(proj A)ǫ,R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Define idempotents of A by  e+  ǫ :=  �  ǫi=1  ei and e−  ǫ :=  �  ǫi=−1  ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We denote by Aǫ the subalgebra of A given by  Aǫ :=  � e+  ǫ Ae+  ǫ  e+  ǫ Ae−  ǫ  0  e−  ǫ Ae−  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  12  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  Define an ideal Iǫ of Aǫ by  Iǫ :=  � rad(e+  ǫ Ae+  ǫ ) ∩ Anne+  ǫ Ae+  ǫ (e+  ǫ Ae−  ǫ )  0  0  rad(e−  ǫ Ae−  ǫ ) ∩ Ann(e+  ǫ Ae−  ǫ )e−  ǫ Ae−  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following result is often very useful to calculate Σǫ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' [AHIKM1, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='26] For each ideal I of Aǫ contained in Iǫ, the isomor-  phisms − ⊗Aǫ A : K0(proj Aǫ)R ≃ K0(proj A)R and − ⊗Aǫ (Aǫ/I) : K0(proj Aǫ)R ≃ K0(proj Aǫ/I)R  gives an isomorphism of fans  Σǫ(A) ≃ Σǫ(Aǫ/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following finiteness condition plays a central role in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a finite dimensional algebra over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We say that A is g-finite if  #2-siltA < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (This is called τ-tilting finite in [DIJ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=')  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A is g-finite (or equivalently, Σ(A) is finite) if and only if Σ(A) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Silting complexes in terms of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we give basic properties of  2-term presilting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Throughout this subsection, we assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For rings A and B and an Aop ⊗k B-module X which is finitely generated on  both sides, let  Λ :=  � A  X  0  B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Equivalently, Λ is a ring with orthogonal idempotents 1 = e1 + e2 satisfying e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In fact,  we can recover Λ from A := e1Λe1, B := e2Λe2 and X := e1Λe2 by the equality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Consider projective Λ-modules  P1 := [A X], P2 := [0 B] ∈ proj Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For s, t ≥ 0, we denote by Ms,t(X) the set of s × t matrices with entries in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have an  isomorphism  Ms,t(X) ≃ HomΛ(P ⊕t  2 , P ⊕s  1 )  sending x ∈ Ms,t(X) to the left multiplication x(·) : P ⊕t  2  → P ⊕s  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus we have a 2-term complex  Px := (P ⊕t  2  x(·)  −−→ P ⊕s  1 ) ∈ per Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following observation is basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let s, t, u, v ≥ 0, x ∈ Ms,t(X) and y ∈ Mu,v(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) Then we have an exact sequence  Mu,s(A) ⊕ Mv,t(B)  [(·)x y(·)]  −−−−−−→ Mu,t(X) → Homper Λ(Px, Py[1]) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) In particular, Px is presilting if and only if Ms,t(X) = Ms(A)x + xMt(B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The assertion (a) follows from an exact sequence  HomΛ(P ⊕s  1  , P ⊕u  1  ) ⊕ HomΛ(P ⊕t  2 , P ⊕v  2  )  [(·)x y(·)]  −−−−−−→ HomΛ(P ⊕t  2 , P ⊕u  1  ) → Homper Λ(Px, Py[1]) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The assertion (b) is immediate from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  The following construction of silting complexes of Λ will be used frequently, where t(XB) (re-  spectively, t(AX)) is the minimal number of generators of X as a right B-module (respectively,  left A-module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10, assume that A and B are local k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) Σ(Λ) contains cone{(0, −1), (1, −r)} for r := t(XB) = dim(X/XJB)B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Σ(Λ) contains cone{(1, 0), (ℓ, −1)} for ℓ := t(AX) = dimA/JA(X/JAX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  13  (c) Let g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , gr be a minimal set of generators of the B-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then µ+  1 (Λ[1]) = Pg ⊕P2[1] ∈  2-siltΛ holds for g := [g1 · · · gr] ∈ M1,r(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (d) Let h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , hℓ a minimal set of generators of the Aop-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then µ−  2 (Λ) = P1 ⊕ Ph ∈  2-siltΛ holds for h :=  � h1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  hℓ  �  ∈ Mℓ,1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12, a part of Σ(Λ) has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(Λ) = P2  P1  Ph=ℓP1−P2  Pg=P1−rP2  P2[1]  P1[1]  +  −  µ−  2 (Λ)  µ+  1 (Λ[1])  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄❄❄❄❄❄  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ✯✯✯✯✯✯✯✯✯✯✯✯✯  ✴✴✴✴✴✴✴✴✴  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We only prove (a)(c) since (b)(d) are the duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A minimal right (add P2[1])-approximation  of P1[1] is given by  g(·) : P2[1]⊕r → P1[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus the mutation of Λ[1] at P1[1] is Pg ⊕ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Now we assume that B is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We fix a minimal set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , gr of the  right B-module X and set  g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB),  where (·) is a canonical surjection X ։ X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have an isomorphism  g(·) : Mr,1(B/JB) ≃ X/XJB,  and we define a map π : X → Mr,1(B/JB) by  π := (X  (·)  −→ X/XJB  (g(·))−1  −−−−−→ Mr,1(B/JB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For each s, t ≥ 0, an entry-wise application of π gives a map  π : Ms,t(X) → Ms,t(Mr,1(B/JB)) = Mrs,t(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In other words, for the identity matrix Is ∈ Ms(k) and gIs :=  � g  O  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  O  g  �  ∈ Ms(M1,r(k)) =  Ms,rs(k), we have  x = (gIs)π(x) for each x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  Define a morphism of k-algebras  φ : Ms(A) → Mrs(B/JB) by a(gIs) = (gIs)φ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Later we will use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10, assume that B is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) π : Ms,t(X) → Mrs,t(B/JB) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard  Mrs,t(B/JB) as an Ms(A)op-module via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Let x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If Px is presilting, then π(x) ∈ Mrs,t(B/JB) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (a) For any a ∈ Ms(A), x ∈ Ms,t(X) and b ∈ Mt(B), we need to show π(axb) = φ(a)π(x)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In fact,  (gIs)φ(a)π(x)b = a(gIs)π(x)b  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  = axb = axb  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  = (gIs)π(axb)  gives the desired equality since gIs(·) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  14  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  (b) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b), we have Ms,t(X) = Ms(A)x + xMt(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Applying π, we have  Mrs,t(B/JB) = π(Ms(A)x + xMt(B))  (a)  =  φ(Ms(A))π(x) + π(x)Mt(B)  ⊂  Mrs(B/JB)π(x) + π(x)Mt(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus the right-hand side is Mrs,t(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This clearly implies that π(x) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  For completeness, we also give the dual statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Now we assume that A  is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We fix a minimal set of generators h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , hℓ of the left A-module X and set  h :=  � h1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  hℓ  �  ∈ Mℓ,1(X) and h :=  � h1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  hℓ  �  ∈ Mℓ,1(X/JAX),  where by abuse of notations, (·) is a canonical surjection X ։ X/JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have an isomor-  phism (·)h : M1,ℓ(A/JA) ≃ X/JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By abuse of notations, let  π := (X  (·)  −→ X/JAX  ((·)h)−1  −−−−−→ M1,ℓ(A/JA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For each s, t ≥ 0, an entry-wise application of π gives a map  π : Ms,t(X) → Ms,t(M1,ℓ(A/JA)) = Ms,ℓt(A/JA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Define a morphism of k-algebras  φ : Mt(B) → Mℓt(A/JA) by (hIt)b = φ(b)(hIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We have the following dual of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10, assume that A is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) π : Ms,t(X) → Ms,ℓt(A/JA) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard  Ms,ℓt(A/JA) as an Mt(B)-module via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Let x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If Px is presilting, then π(x) ∈ Ms,ℓt(A/JA) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Uniserial property of g-finite algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' As an application of results in the previous sub-  section, we prove the following result, which is not used in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be a finite dimensional elementary k-algebra, and 1 = e1 + · · · + en the  orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If Λ is g-finite, then for each 1 ≤ i ̸= j ≤ n, eiΛej/eiΛejJΛej  is a uniserial (eiΛei)op-module and eiΛej/eiJΛeiJΛej is a uniserial ejΛej-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thanks to sign decomposition, we can deduce Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15 from the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A and B be local k-algebras with k ≃ A/JA ≃ B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If X is a Aop ⊗k B-  module such that  �  A  X  0  B  �  is g-finite, then X/XJB is a uniserial Aop-module and X/JAX is a  uniserial B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16⇒Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Λ is g-finite, so is Γ := (ei + ej)Λ(ei + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7, Γ+− =  � eiΛei  eiΛej  0  ejΛej  �  is also g-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus the assertion follows from Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  In the rest of this subsection, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following observation plays a key role in the proof, where we identify K0(proj Λ) with Z2  via [A X] �→ (1, 0), [0 B] �→ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ :=  � A  X  0  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume that (1, −1) ∈ K0(proj Λ) is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) There exists h ∈ X such that X = Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  15  (b) Let Λ′ :=  � A  JAX  0  k  �  and t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If (1, −t) ∈ K0(proj Λ) is rigid, then (1, 1 − t) ∈ K0(proj Λ′)  is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (a) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b), there exists h ∈ X satisfying X = Ah + hk = Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b), there exists [x1 x2 · · · xt] ∈ M1,t(X) such that  M1,t(X) = A[x1 · · · xt] + [x1 · · · xt]Mt(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2)  As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2, the element h gives surjections  π := (X  (·)  −→ X/JAX  ((·)h)−1  −−−−−→ A/JA = k) and π : M1,t(X) → M1,t(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='14, π(x) ∈ M1,t(k) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By changing indices if necessary, we can assume  x1 ∈ A×h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Multiplying an element in A× from left, we can assume x1 = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Multiplying an element  in GLt(k) from right, we can assume xi ∈ JAh for each 2 ≤ i ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We claim  M1,t−1(JAX) = A[x2 · · · xt] + [x2 · · · xt]Mt−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In fact, fix any [y2 · · · yt] ∈ M1,t−1(JAX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2) there exist a ∈ A and b = [bij]1≤i,j≤t ∈ Mt(k)  such that  [0 y2 · · · yt] = a[h x2 · · · xt] + [h x2 · · · xt]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3)  Applying π, we obtain  [0 0 · · · 0] = a[1 0 · · · 0] + [1 0 · · · 0]b in M1,t(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus we obtain b12 = · · · = b1t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Looking at the i-th entries for 2 ≤ i ≤ t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3), we have  [y2 · · · yt] = a[x2 · · · xt] + [x2 · · · xt][bij]2≤i,j≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We are ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We prove that X/XJB is a uniserial Aop-module under a weaker assump-  tion that (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Λ :=  � A  X/XJB  0  k  �  is a factor  algebra of Λ, the element (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Replacing Λ by Λ, we can  assume that  B = k and Λ =  � A  X  0  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We use induction on dimk X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='17(a), the Aop-module X has a unique maximal  submodule JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ′ =  � A  JAX  0  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='17(b), (1, −t) ∈ K0(proj Λ′) is rigid for  each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By induction hypothesis, JAX is a uniserial Aop-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Therefore X is also a  uniserial Aop-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Gluing, Rotation and Subdivision of g-fans  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Gluing fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive  idempotents 1 = e1 + e2 ∈ Λ and 1 = e′  1 + e′  2 ∈ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we prove the following  Gluing Theorem, where we identify K0(proj Λ) and K0(proj Λ′) with Z2 by e1Λ = (1, 0) = e′  1Λ′ and  e2Λ = (0, 1) = e′  2Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1 (Gluing Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal  primitive idempotents 1 = e1 + e2 ∈ Λ and 1 = e′  1 + e′  2 ∈ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume e1Λe2 = 0 and e′  2Λ′e′  1 = 0,  or equivalently, Σ(Λ) ∈ Fan  +−  sc (2) and Σ(Λ′) ∈ Fan  −+  sc (2) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then, there exists an  elementary k-algebra Γ such that  Σ(Γ) = Σ(Λ) ∗ Σ(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  16  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(Λ) = +  −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Σ(Λ′) = +  −  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  Σ(Γ) = +  −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  The construction of Γ is as follows: We can write  Λ =  �  A  X  0  B  �  and Λ′ =  �  C  0  Y  D  �  ,  where A, B, C, D are local k-algebras, X is an Aop ⊗k B-module, and Y is an Dop ⊗k C-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since Λ and Λ′ are elementary, we have k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A ×k C be a  fiber product of canonical surjections (·) : A → k and (·) : C → k, that is,  A ×k C := {(a, c) ∈ A × C | a = c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Let B ×k D be a fibre product of (·) : B → k and (·) : D → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Using the projections A ×k C → A  and B ×k D → B, we regard X as an (A ×k C)op ⊗k (B ×k D)-module, and using the projections  A ×k C → C and B ×k D → D, we regard Y as an (B ×k D)op ⊗k (A ×k C)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We prove that the algebra  Γ :=  � A ×k C  X  Y  B ×k D  �  satisfies Σ(Γ) = Σ(Λ) ∗ Σ(Λ′), where the multiplication of the elements of X and those of Y are  defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' It suffices to prove  Σ+−(Γ) = Σ+−(Λ) and Σ−+(Γ) = Σ−+(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For ǫ = (+, −), we have Γǫ =  �  A ×k C  X  0  B ×k D  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The ideal I :=  �  rad C  0  0  rad D  �  of Γǫ is  contained in Iǫ, and we have an isomorphism Γǫ/I ≃ Λ of k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 to  Γ, we get Σ+−(Γ) = Σ+−(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By the same argument, Σ−+(Γ) = Σ−+(Λ′) holds, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ and Λ′ be the following algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ :=  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a3 �  a4  �  a2  �  a1  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨a2  1, a2  2, a2  4, a2a1, a2a3 − a3a4⟩,  Λ′ :=  k  �  1  2  b1  �  b2  �  �  ⟨b2  2⟩  By Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11 below, we have  Σ(Λ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='00 =  Σ(Λ′) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1212 =  Let A = e1Λe1, X = e1Λe2, B = e2Λe2, C = e1Λ′e1, Y = e2Λ′e1, D = e2Λ′e2 and Γ =  � A ×k C  X  Y  B ×k D  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then  Γ =  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a3 �  a4  �  a2  �  a1  � b1  �  b2  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨a2  1, a2  2, a2  4, a2a1, a2a3 − a3a4, b2  2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  17  By Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1, we have  Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1212 =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Rotation and Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we explain a connection between the rotation  of a fan given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13 and mutation of a 2-term silting complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following main result in this section shows that mutation of an algebra induce the rotation  of the g-fan, where we identify K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3 (Rotation Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be a finite dimensional k-algebra of rank 2 with or-  thogonal primitive idempotents 1 = e1 + e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan  +−  sc (2)  (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then, there exists a finite dimensional k-algebra Γ such that  Σ(Γ) = ρ(Σ(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Furthermore, if Λ is elementary, then Γ can be taken to be elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(Λ) = +  −  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄❄❄❄❄❄  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  Σ(Γ) ≃ +  −❄❄❄❄❄❄  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❚  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  ❲  To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3, we need the following preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Let A be a basic finite dimensional algebra over a field k with |A| = n, and 1 = e1 + · · · + en  the orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For 1 ≤ i ≤ n and δ ∈ {±1}, consider a half space  Rn  i,δ := {x1e1 + · · · + xden ∈ Rn | δxi ≥ 0}  and define a subfan of Σ by  Σi,δ := {σ ∈ Σ | σ ⊂ Rn  i,δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  On the other hand, for elements T ≥ T ′ in siltA, we consider the interval  [T ′, T ] := {U ∈ siltA | T ≥ U ≥ T ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following result provides a correspondence of a part of two g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For 1 ≤ i ≤ n, let B := EndA(µ−  i (A)), where µ−  i (A) = Ti ⊕ (�  j̸=i P A  j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) [AHIKM1, Threom 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='26] There exists a triangle functor F : Kb(proj A) → Kb(proj B) which  satisfies F(Ti) ≃ P B  i  and F(P A  j ) ≃ P B  j  for each j ̸= i and gives an isomorphism K0(proj A) ≃  K0(proj B) and an isomorphism of fans  Σi,−(A) ≃ Σi,+(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) There are isomorphisms (1 − ei)A(1 − ei) ≃ (1 − ei)B(1 − ei) and A/(1 − ei) ≃ B/(1 − ei) of  k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (b) Although this is known to experts, we give a proof for convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The first  isomorphism is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' To prove the second one, notice that A/(1 − ei) = EndKb(proj A)(P A  i )/[A/P A  i ]  and B/(1−ei) = EndKb(proj A)(Ti)/[T/Ti] hold, where [X] denotes the ideal consisting of morphisms  factoring through add X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Pi  f−→ Q  g−→ Ti  h−→ Pi[1] be an exchange triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let a ∈ eiAei =  EndKb(proj A)(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since f is a minimal left (add A/Pi)-approximation of Pi, we obtain the following  commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Pi  f  �  a�  Q  g  �  �  Ti  h �  b�  Pi[1]  a[1]  �  Pi  f  � Q  g  � Ti  h � Pi[1]  18  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  It is routine to check that the desired isomorphism A/(1 − ei)A = EndKb(proj A)(P A  i )/[A/P A  i ] ≃  B/(1 − ei) = EndKb(proj A)(Ti)/[T/Ti] is given by a �→ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let T = P Λ  1 ⊕ T2 := µ−  2 (Λ) and E := EndKb(proj Λ)(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4(a), we have a triangle functor F : Kb(proj Λ) → Kb(proj E) which satisfies  F(P Λ  1 ) = P E  1  and F(T2) = P E  2  and induces an isomorphism F : K0(proj Λ) ≃ K0(proj E) and an isomorphism of fans  F : Σ2,−(Λ) ≃ Σ2,+(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Σ(Λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='P Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='P Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='T2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='❄❄❄❄❄❄  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='❚ Σ(E)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='P E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='❲  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❲  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❲  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❲  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 to E, we obtain a k-algebra Γ := E−+ such that  e1Γe2 = 0 and Σ−+(Γ) = Σ−+(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Therefore under the isomorphism K0(proj Γ) ≃ Z2 given by P Γ  1 �→ (0, 1) and P Γ  2 �→ (1, 0), we have  Σ(Γ) = ρ(Σ(Λ)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  It remains to prove the last assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4(a), we have isomorphisms e1Ee1 ≃  e1Λe1 and Λ/(e1) ≃ E/(e1) of k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus, if Λ is elementary, then so are E and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We give two examples of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The first one satisfies E = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ =  k  �  1  2  a �  b  �  �  ⟨b2⟩  Σ(Λ) = Σ1212 =  We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ  a·  −→ e1Λ] ⊕ e1Λ and E := EndKb(proj Λ)(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have  Γ = E =  k  �  1  2  a �  b  �  �  ⟨b2⟩  and Σ(Γ) = ρ(Σ(Λ)) = Σ2121 =  The second example satisfies E ̸= Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ =  k  �  1  2  a �  b  �  c  �  �  ⟨b2, c2, bac⟩  Σ(Λ) = Σ21312 =  We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ ( a·  ac·)  −−−−→ e1Λ⊕2] ⊕ e1Λ and E := EndKb(proj Λ)(T ), where we  switch the indices 1 and 2 unlike the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then  E =  k  �  1  2  a �  a′  �  b  �  �  ⟨b2, a′b, a′aa′⟩  and Σ(E) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='111 =  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  19  where new arrows a, a′ and b are morphisms in Kb(proj Λ) given by commutative diagrams  0  e1Λ  e2Λ  e1Λ⊕2  �  �  ( a·  ac·) �  ( 0  1)  �  e2Λ  e1Λ⊕2  0  e1Λ  ( a·  ac·) �  �  �  ( 0 b· )  �  e2Λ  e1Λ⊕2  e2Λ  e1Λ⊕2  ( a·  ac·) �  c· �  ( a·  ac·) �  ( 0 1  0 0)  �  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Γ := E+− =  � e1Ee1  e1Ee2  0  e2Ee2  �  =  � ⟨e1, b, aa′, baa′⟩k  ⟨a, ba, aa′a, baa′a⟩k  0  ⟨e2, a′a⟩k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then  Γ =  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a �  c  �  b  �  b′  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨b2, b′2, c2, b′b, b′a − ac⟩ and Σ(Γ) = ρ(Σ(Λ)) = Σ13122 =  where b′ := aa′ and c := a′a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Subdivision and Extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this section, we realize subdivisions of g-fans of rank 2 by  extensions of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The following theorem is a main result of this section, where we identify  K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 (Subdivision Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be a finite dimensional elementary k-algebra with  orthogonal primitive idempotents 1 = e1+e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan  +−  sc (2)  (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then, for cones σ = C(µ+  1 (Λ[1])) and σ′ := C(µ−  2 (Λ)) of Σ(Λ), there exist finite  dimensional elementary k-algebras Γ and Γ′ such that  Σ(Γ) = Dσ(Σ(Λ)) and Σ(Γ′) = Dσ′(Σ(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='Σ(Λ) = P2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='P1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='P2[1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='µ−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2 (Λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='µ+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1 (Λ[1])  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='✯✯✯✯✯✯✯✯✯✯✯✯✯  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='❚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='❚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='✴✴✴✴✴✴✴✴✴  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='In the rest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' we only prove the existence of Γ since the existence of Γ′ is the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The construction of Γ is as follows:  Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5, we can write  Λ =  � A  X  0  B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  where A, B are local k-algebras and X is an Aop ⊗k B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Λ is elementary, we have  k ≃ A/JA ≃ B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let  X := X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then the k-dual DX is an A-module, and we regard it as an Aop-module by using the action of k  through the natural surjection A → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let  C := A ⊕ DX  be a trivial extension algebra of A by DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let  (·) : A → k, (·) : B → k and (·) : X → X  20  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  be canonical surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We regard  Y :=  � k  X  �  as a Cop ⊗k B-module by  (a, f) · [ α  x ] · b :=  �  aαb+f(x)b  axb  �  for (a, f) ∈ C = A ⊕ DX, [ α  x ] ∈ Y =  � k  X  �  and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then we set  Γ :=  �  C  Y  0  B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In the rest of this subsection, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We set  Q1 := [C Y ], Q2 := [0 B] ∈ proj Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For y ∈ Ms,t(Y ) ≃ HomΓ(Q⊕t  2 , Q⊕s  1 ), we define  Qy := [Q⊕t  2  y(·)  −−→ Q⊕s  1 ] ∈ Kb(proj Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We fix a minimal set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , gr of the B-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , gr) forms a  k-basis of X = X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Set  g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We need the following easy observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Σ(Γ) contains cone{(0, 1), (1, −r−1)} and cone{(1, −r−1), (1, −r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' More explicitly,  let  � 0  g  �  ∈ M1,r(Y ) and  � 0 1  g 0  �  ∈ M1,r+1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then Q� 0 1  g 0  � ⊕ Q2[1] and Q� 0  g  � ⊕ Q� 0 1  g 0  � belong to 2-siltΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A minimal set of generators of the B-module Y is given by the r+1 columns of  � 0 1  g 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus  Q� 0 1  g 0  � ⊕ Q2[1] ∈ 2-siltΓ holds by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In the rest, we prove that T := Q� 0  g  � ⊕ Q� 0 1  g 0  � is basic silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By the first statement, Q� 0 1  g 0  �  is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If Q� 0  g  � is not indecomposable, then |T | is bigger than two, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus T is basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We will show that T is presilting by using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By our choice of g, we have  gMr,1(B) = X and (DX)g = M1,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus we have  � 0  g  �  Mr(B) = M1,r([ 0  X ]) and (DX)  � 0  g  �  = M1,r([ k  0 ]), and hence  C  � 0  g  �  +  � 0  g  �  Mr(B) ⊃ (DX)  � 0  g  �  +  � 0  g  �  Mr(B) = M1,r([ 0  X ]) + M1,r([ k  0 ]) = M1,r(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  This clearly implies  C  � 0  g  �  +  � 0 1  g 0  �  Mr+1,r(B) = M1,r(Y ),  and a similar argument implies  C  � 0 1  g 0  �  +  � 0  g  �  Mr,r+1(B) = M1,r+1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b) implies that T is presilting, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2, the element g gives a surjection  π := (X  (·)  −→ X  (g(·))−1  −−−−−→ Mr,1(B) = Mr,1(k)),  which extends to the map π : Ms,t(X) → Mrs,t(k) for each s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following observation is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For x ∈ Ms,t(X), consider [ 0  x ] ∈ Ms,t(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  21  (a) Px is indecomposable in Kb(proj Λ) if and only if Q[ 0  x] is indecomposable in Kb(proj Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) If Q[ 0  x] is a presilting complex of Γ, then Px is a presilting complex of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) The converse of (b) holds if t ≤ rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (d) The restriction of Σ(Γ) to {(x, y) ∈ R2 | 0 ≤ −y ≤ rx} coincides with that of Σ(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Notice that Γ is the trivial extension Λ ⊕ I of Λ by the following ideal I of Γ:  I :=  �  DX  k  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) Since Px ≃ Q[ 0  x] ⊗Γ Λ and Q[ 0  x] ≃ Px ⊗Λ Γ, the assertion follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Since Λ = Γ/I and Q[ 0  x] ⊗Γ Λ ≃ Px, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (c) Assume that Px is a presilting complex of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b), we have  Ms,t(X) = Ms(A)x + xMt(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2)  Again by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11(b), it suffices to show the equality  V := Ms(C) [ 0  x ] + [ 0  x ] Mt(B) = Ms,t(  � k  X  �  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since V ⊃ Ms(A) [ 0  x ] + [ 0  x ] Mt(B)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2)  = Ms,t([ 0  X ]) holds, it suffices to show  V ⊃ Ms,t([ k  0 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3)  By our assumption t ≤ rs and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13(b), π(x) has rank t and the map  (·)π(x) : Ms,rs(k) → Ms,t(k)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4)  is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We denote by g∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , g∗  r the basis of DX which is dual to g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then the map  (·)  � g∗  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  g∗  r  �  : M1,r(k) ≃ DX is a bijection, and we denote its inverse by  π′ : DX ≃ M1,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  It gives a bijection π′ : Ms(DX) ≃ Ms,rs(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We have a commutative diagram  Ms(DX) × Ms,t(X)  π′×π  �  eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  �❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  ❱  Ms,rs(k) × Mrs,t(k)  mult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  �  Ms,t(k)  where eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' is given by the evaluation map DX × X → DX × X → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus the commutativity of  the diagram above and the surjectivity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4) shows that the map  (·)x : Ms(DX) → Ms,t(k)  is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Therefore the desired claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3) follows from  V ⊃ Ms(C) [ 0  x ] ⊃ Ms(DX) [ 0  x ] = Ms,t([ k  0 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  (d) Immediate from (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The assertion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We give two examples of Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  22  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Σ(Λ) is the following fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ = k[1 → 2]  Σ(Λ) = Σ111 =  Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 to Λ, we get  Γ :=  �  k ⊕ Dk  � k  k  �  0  k  �  =  k  �  1  2  �  b  �  �  ⟨b2⟩  and Σ(Γ) = D3(Σ(Λ)) = Σ1212 =  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ =  k  �  1  2  a �  b  �  �  ⟨b2⟩  Σ(Λ) = Σ2121 =  Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 to Λ, we get  Γ :=  �  k ⊕ D(ka)  �  k  ⟨a,ab⟩k  �  0  ⟨e2, b⟩k  �  =  k  �  1  2  a �  c  �  b  �  �  ⟨b2, c2, cab⟩  and Σ(Γ) = D4(Σ(Λ)) = Σ21312 =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let k be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For a finite dimensional k-algebras Λ of rank 2,  we regard the g-fan Σ(Λ) as a fan in R2 by isomorphism K0(proj Λ) ≃ R2 given by P1 �→ (1, 0) and  P2 �→ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We denote by  k-Fan(2)  the subset of Fansc(2) consisting of g-fans of finite dimensional k-algebras of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let k-Fanel(2)  be the subset of k-Fan(2) consisting of g-fans of finite dimensional elementary k-algebras of rank  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following is a main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For any field k, we have  k-Fanel(2) = k-Fan(2) = Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5)  That is, any sign-coherent fan in R2 can be realized as a g-fan Σ(Λ) of some finite dimensional  elementary k-algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' It suffices to show Fansc(2) = k-Fanel(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let  k-Fan+−  el (2) := k-Fanel(2) ∩ Fan+−  sc (2) and k-Fan−+  el (2) := k-Fanel(2) ∩ Fan−+  sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1, we have  k-Fanel(2) = k-Fan+−  el (2) ∗ k-Fan−+  el (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3, k-Fan+−  el (2) is closed under rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 and Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='14(b), k-Fan+−  el (2) is closed under subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) = Σ(k × k) ∈ k-Fan+−  el (2),  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10 implies  k-Fan+−  el (2) = Fan+−  sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Similarly, we have k-Fan−+  el (2) = Fan−+  sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Consequently, we have  Fansc(2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5)  = Fan+−  sc (2) ∗ Fan−+  sc (2) = k-Fan+−  el (2) ∗ k-Fan−+  el (2) = k-Fanel(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  For given Σ ∈ Fansc(2), our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13 gives a concrete algorithm to construct a  finite dimensional k-algebra Λ satisfying Σ(Λ) = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We demonstrate it in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We construct a finite dimensional k-algebra Γ satisfying Σ(Γ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1212 by  the following three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  23  (I) We obtain a finite dimensional k-algebra  Λ =  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a3 �  a4  �  a2  �  a1  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨a2  1, a2  2, a2  4, a2a1, a2a3 − a3a4⟩  satisfying Σ(Λ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='00 by using Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3 and Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7 as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ00  D2  �  Σ111  D3  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11  �  Σ1212  ρ  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5  �  Σ2121  D4  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12  �  Σ21312  ρ  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6  �  Σ13122  (II) Similarly, we obtain a finite dimensional k-algebra  Λ′ :=  k  �  1  2  b1  �  b2  �  �  ⟨b2  2⟩  satisfying Σ(Λ′) = Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 1, 2, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (III) We obtain a finite dimensional k-algebra  Γ =  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a3 �  a4  �  a2  �  a1  � b1  �  b2  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨a2  1, a2  2, a2  4, a2a1, a2a3 − a3a4, b2  2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩  satisfying Σ(Γ) = Σ(1, 3, 1, 2, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 1, 2, 1, 2) by applying Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1 to Λ and Λ′, see  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(Λ) =  Σ(Λ′) =  Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Gluing fans II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we study another type of gluing g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Results in this  subsection will not be used in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idem-  potents 1 = e1 + e2 ∈ Λ and 1 = e′  1 + e′  2 ∈ Λ′ satisfying e1Λe2 = 0, e′  1Λ′e′  2 = 0,  σ = cone{(0, −1), (1, −1)} ∈ Σ(Λ)  and σ′ = cone{(1, −1), (1, 0)} ∈ Σ(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6)  Then, there exists an elementary k-algebra Γ such that  Σ2(Γ) = (Σ2(Λ) \\ {σ}) ∪ (Σ2(Λ′) \\ {σ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(Λ) = +  −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  P1  P2  σ  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  Σ(Λ′) = +  −  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  P ′  1  P ′  2  σ′  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  Σ(Γ) = +  −  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Q1  Q2  ❄❄❄❄❄❄  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  24  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  The assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6) is equivalent to that the defining sequences can be written as  Σ(Λ) = Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0) and Σ(Λ′) = Σ(1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  In this case, the defining sequence of Σ(Γ) is given by  Σ(Γ) = Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an−2, an−1 + b2 − 1, b3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The rest of this section is devoted to proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By our assumption, we can write  Λ =  �  A  X  0  B  �  and P1 := [A X], P2 := [0 B] ∈ proj Λ,  Λ′ =  � C  Y  0  D  �  and P ′  1 := [C Y ], P ′  2 := [0 D] ∈ proj Λ′,  where  A, B, C, D are local k-algebras such that k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  X is an Aop ⊗k B-module and Y is an Cop ⊗k D-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  There exist g ∈ X and h ∈ Y such that X = gB ̸= 0 and Y = Ch ̸= 0 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The construction of Γ is as follows: Let A ×k C (respectively, B ×k D) be a fibre product of  canonical surjections A → k and C → k (respectively, B → k and D → k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2, we  consider maps  π : X → X/XJB  (g(·))−1  −−−−−→ B/JB = k and π′ : Y → Y/JCY  ((·)h)−1  −−−−−→ C/JC = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7)  Let X×kY be a fibre product of π : X → k and π′ : Y → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then X×kY is a (A×kC)op⊗k(B×kD)-  module, and let  Γ :=  � A ×k C  X ×k Y  0  B ×k D  �  and Q1 := [A ×k C X ×k Y ], Q2 := [0 B ×k D] ∈ proj Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Consider ideals of Γ by  I =  � JC  JCY  0  JD  �  and I′ =  � JA  XJB  0  JB  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then there exist isomorphisms of k-algebras  Γ/I ≃ Λ and Γ/I′ ≃ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='8)  As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2, for s, t ≥ 0, x ∈ Ms,t(X), y ∈ Ms,t(Y ) and (x′, y′) ∈ Ms,t(X ×k Y ), we define  Px  :=  (P ⊕t  2  x(·)  −−→ P ⊕s  1 ) ∈ per Λ,  P ′  y  :=  (P ′  2  ⊕t  y(·)  −−→ P ′  1  ⊕s) ∈ per Λ′  Q(x,y)  :=  (Q⊕t  2  (x′,y′)(·)  −−−−−−→ Q⊕s  1 ) ∈ per Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let s, t ≥ 0 and (x, y) ∈ Ms,t(X ×k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' If Q(x,y) is a presilting complex of Γ,  then Px is a presilting complex of Λ and P ′  y is a presilting complex of Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='8) and Q(x,y) ⊗Γ Λ = Px, the complex Px is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The complex P ′  y is presilting  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Define maps (·) : A → C and (·) : B → D as the compositions of canonical maps  (·) : A  (·)  −→ k ⊂ C and (·) : B  (·)  −→ k ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Using π and π′ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='7), define maps (·) : X → Y and (·) : Y → X by  (·) : X  π−→ k  (·)h  −−→ kh ⊂ Y  and (·) : Y  π′  −→ k  (·)g  −−→ kg ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9)  Then the first projection X ×k Y → X, (x, y) �→ x has a section given by  X → X ×k Y, x �→ (x, x),  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  25  and the second projection X ×k Y → Y , (x, y) �→ y has a section given by  Y → X ×k Y, y �→ (y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The following is a crucial result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) Let s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For x ∈ Ms,t(X), consider (x, x) ∈ Ms,t(X ⊗k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Px is a presilting complex  of Λ if and only if Q(x,x) is a presilting complex of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Let s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For y ∈ Ms,t(Y ), consider (y, y) ∈ Mc,d(X ⊗k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then P ′  y is a presilting complex  of Λ′ if and only if Q(y,y) is a presilting complex of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' It suffices to prove (a) since (b) is dual to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The “if” part is clear from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11, it suffices to show  Ms,t(X ×k Y ) = Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since  X ×k Y = {(0, y) | y ∈ JCY } + {(z, z) | z ∈ X},  it suffices to show the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (i) For each y ∈ Ms,t(JCY ), we have (0, y) ∈ Ms(A ×k C)(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (ii) For each z ∈ Ms,t(X), we have (z, z) ∈ Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Px is presilting, π(x) ∈ Ms,t(k) has full rank by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since s ≥ t,  the map (·)π(x) : Ms(k) → Ms,t(k) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Applying JC ⊗k −, the map (·)π(x) : Ms(JC) →  Ms,t(JC) is also surjective, and so is the composition  (·)x  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='9)  = (·)π(x)h : Ms(JC)  (·)π(x)  −−−−→ Ms,t(JC)  (·)h  −−→ Ms,t(JCY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Therefore there exists c ∈ Ms(JC) such that y = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then (0, c) ∈ Ms(A ×k C) satisfies  (0, c)(x, x) = (0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  We prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Px is presilting, we have Ms,t(X) = Ms(A)x + xMt(B) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus there exist a ∈ Ms(A) and b ∈ Mt(B) such that z = ax + xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then  (a, a)(x, x) + (x, x)(b, b) = (ax + xb, ax + xb) = (z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Now we are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5, each of Σ(Λ), Σ(Λ′) and Σ(Γ) contains cone{(−1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='16 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='17, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (i) Let s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then there exists x ∈ Ms,t(X) such that Px is presilting if and only if there exists  (x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (ii) Let s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then there exists y ∈ Ms,t(Y ) such that P ′  y is presilting if and only if there exists  (x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Therefore the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let Λ and Λ′ be the following algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Λ :=  k  �  1  2  a �  b  �  �  ⟨b2⟩  Λ′ :=  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a �  d  �  c1  �  c2  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨c2  1, c2  2, d2, c1c2, c1a − ad⟩  26  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  By Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6, we have  Σ(Λ) = Σ2121 =  Σ(Λ′) = Σ13122 =  Let Γ :=  � e1Λe1 ×k e1Λ′e1  e1Λe2 ×k e1Λ′e2  0  e2Λe2 ×k e2Λ′e2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have  Γ =  k  \uf8ee  \uf8ef\uf8ef\uf8f0  1  2  a �  b  �  d  �  c1  �  c2  �  \uf8f9  \uf8fa\uf8fa\uf8fb  ⟨b2, c2  1, c2  2, d2, c1c2, c1a − ad, c2ab, bd, db⟩ and Σ(Γ) = Σ214122 =  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' g-Convex algebras of rank 2  In this section, we will characterize algebras of rank 2 which have convex g-polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Characterizations of g-convex algebras of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let e, e′ be pairwise orthogonal prim-  itive idempotents in A and x ∈ eAe′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  x ∈ eAe′ is a left generator (respectively, right generator) of eAe′ if eAx = eAe′ (respectively,  xAe′ = eAe′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Define subalgebras Lx ⊂ e′Ae′ and Rx ⊂ eAe as follows (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Rx := {a ∈ eAe | ax ∈ xAe′} and Lx := {a ∈ e′Ae′ | xa ∈ eAx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Recall that, for an algebra Λ and a right (respectively, left) Λ-module M, we denote by t(MΛ)  (respectively, t(ΛM)) the minimal number of generators of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be a basic finite dimensional algebra, {e1, e2} a complete set of primitive  orthogonal idempotents in A and Pi = eiA (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (a) A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some  left generator x of e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ(A) = Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some  right generator x of e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ⑧⑧⑧⑧⑧⑧  Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❄  ❄  ❄  ❄  ❄  ❄  ✴✴✴✴✴✴✴✴✴  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b P2  P1  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' For a left (respectively, right) generator x of e1Ae2, Rx (respectively, Lx) is unique  up to conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In particular, t(Rxe1Ae1) (respectively, t(e2Ae2Lx)) does not depend on the choice  of a left (respectively, right) generator x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  27  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' (a) Here, we give algebras which realize 7 convex g-fans up to isomorphism of  g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We define Ai = kQ/I (i ∈ {1, 2, 3, 4, 5, 6, 7}) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  A1 =  k  �  1  2  a �  b�  c  �  d  �  �  ⟨ab, ad, ba, bc, c2, d2⟩  A2 =  k  �  1  2  a �  b�  d  �  �  ⟨ab, ba, d2⟩  A3 =  �  1  2  a �  b�  d  �  �  ⟨ab, ba, ad, d2⟩  A4 =  k  �  1  2  b�  d  �  �  ⟨d2⟩  A5 =  k  �  1  2  a �  b�  �  ⟨ab, ba⟩  A6 = k  �  1  2  b�  �  A7 = k  �  1 2  �  Then the g-fans Σ(Ai) (i ∈ {1, 2, 3, 4, 5, 6, 7}) are given by the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  i  1  2  3  4  5  6  7  Σ(Ai)  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ✴✴✴✴✴✴✴  ❄❄❄❄❄  ❖❖❖❖❖❖❖  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄❄❄❄❄  ❄❄❄❄❄  ❖❖❖❖❖❖❖  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❖❖❖❖❖❖❖  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❖❖❖❖❖❖❖  +  −  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  ❄❄❄❄❄  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  ❄❄❄❄❄  +  −  ❄❄❄❄❄  ⑧⑧⑧⑧⑧  ⑧⑧⑧⑧⑧  ❄❄❄❄❄  (b) Let K/k be a field extension with degree two,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' and A be a k-algebra  �k  K  0  K  �  with e1 =  �1  0  0  0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  e2 =  �0  0  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' We write K = k(t) and set x :=  �0  1  0  0  �  ∈ e1Ae2, u =  �0  0  0  t  �  ∈ e2Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we  have Lx =  �0  0  0  k  �  , u ̸∈ Lx, and the following equations hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  e1Ae2 =  �  0  K  0  0  �  = xAe2 = e1Ax + e1Axu  e2Ae2 =  �0  0  0  K  �  = Lx + uLx  Further, we have e2Ae1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Therefore, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1 implies that Σ(A) has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' P2  P1  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❖  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄❄❄❄❄❄  ⑧⑧⑧⑧⑧⑧  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' In this subsection, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The following observation  shows Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1(a) and gives another proof of [AHIKM1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let A be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b for  some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The “if” part is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Conversely, assume that A is g-convex and Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='b with  a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , an) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' , bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then ai ≤ 2 and bj ≤ 2 hold for each i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Using Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='10, it is easy to check that a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)} holds (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Next we show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Let x ∈ e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then Lx is a subalgebra of e2Ae2, and Rx is a subalgebra of e1Ae1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This is a special case of the following easy fact: Let A, B be rings, M an (A, B)-module,  and x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then {b ∈ B | xb ∈ Ax} is a subring of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Now we give a key observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2, for s, t ≥ 0, x ∈ Ms,t(e1Ae2), we define  Px := (e2A⊕t  x(·)  −−→ e1A⊕s) ∈ Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Assume t(e1Ae1e1Ae2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For a left generator x ∈ e1Ae2, the following  conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  28  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  (1) Σ(A) contains cone{(1, −1), (1, −2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2) t(Rxe1Ae2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (3) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2) holds for some u ∈ e1Ae1 \\ Rx and y := [x ux].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Notice that Px is indecomposable presilting by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (1)⇒(2) If t(Rxe1Ae1) = 1, then e1Ae1 = Rx holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus e1Ae2 = e1Ax ⊂ xAe2 holds, and  thus x is a right generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12, Px ⊕ P2[1] ∈ 2-siltA holds, a contradiction to  cone{(1, −1), (1, −2)} ∈ Σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus it suffices to prove t(Rxe1Ae1) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Since cone{(1, −1), (1, −2)} ∈ Σ(A), there exists y = [x1 x2] ∈ M1,2(e1Ae2) such that Px ⊕ Py  is silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11, we have  M1,2(e1Ae2) = e1Ay + yM2,2(e2Ae2),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1)  M1,2(e1Ae2) = e1Ay + xM1,2(e2Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2)  Looking at the first entry of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1), at least one of x1 and x2 does not belong to rade1Ae1 e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Without loss of generality, assume x1 /∈ rade1Ae1 e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then there exists a ∈ (e1Ae1)× such that  x = ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Py ≃ Pay, we can assume x1 = x by replacing y by ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since x is a left generator,  there exists u ∈ e1Ae1 such that x2 = ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Consequently, we can assume  y = [x ux].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  For each a ∈ e1Ae1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='2) implies that there exist a′ ∈ e1Ae1 and b, b′ ∈ e2Ae2 such that  [0 ax] = a′[x ux] + x[b b′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Then a′ and a−a′u are in Rx, and hence a = a′u+(a−a′u) ∈ Rxu+Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus e1Ae1 = Rx +Rxu  and t(Rxe1Ae1) ≤ 2 hold, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (2)⇒(3) Since t(Rxe1Ae2) = 2 and Rx ̸⊂ radRx e1Ae1, there exists u ∈ e1Ae1 \\ Rx such that  Rxu + Rx = e1Ae1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Multiplying x from the right, we have Rxux + Rxx = e1Ax = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since Rxx ⊂ xAe2, we  have  Rxux + xAe2 = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3)  To prove (3), take any [z w] ∈ M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since x is a left generator, there exists a ∈ e1Ae1  such that z = ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='3), there exist r ∈ Rx and b ∈ e2Ae2 such that w − aux = rux + xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By  definition of Rx, there exists c ∈ e2Ae2 such that rx = xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Then we have  [z w] = (a + r)[x ux] + x[−c b] ∈ e1Ay + xM1,2(e2Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  (3)⇒(1) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='11, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Px is presilting if and only if (i) e1Ax + xAe2 = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Py is presilting if and only if (ii) e1Ay + yM2,2(e2Ae2) = M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  HomKb(proj A)(Px, Py[1]) = 0 if and only if (iii) e1Ax + yM2,1(e2Ae2) = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  HomKb(proj A)(Py, Px[1]) = 0 if and only if (iv) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  It is clear that (iv) implies (ii), and (i) implies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By looking at the first entry of the row vector,  (iv) implies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Our assumption (3) implies that (iv) holds, and hence (i)-(iii) also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Thus Px ⊕ Py is  presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' It remains to show that Py is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Suppose that Py is decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By  considering g-vector, we have that Py ≃ e2A[1] ⊕ Pz for some z ∈ e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Since [Pz] = [Px], we  have Pz ≃ Px by [DIJ, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This shows that e2A[1] ⊕ Px is silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12, we have xAe2 = e1Ae2 and Rx = eAe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' This contradicts u ̸∈ Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  We are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' The first and second statements follow from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='5 and Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2  29  We prove the third statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='12, cone{(1, 0), (1, −1)} ∈ Σ(A) if and only if  t(e1Ae1e1Ae2) = 1, and cone{(0, −1), (1, −2)} ∈ Σ(A) if and only if t(e1Ae2e2Ae2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Thus the  assertion follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  The fourth statement is the dual of the third statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  □  Acknowledgments  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='A is supported by JSPS Grants-in-Aid for Scientific Research JP19J11408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='H is supported  by JSPS Grant-in-Aid for Scientists Research (C) 20K03513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='I is supported by JSPS Grant-  in-Aid for Scientific Research (B) 16H03923, (C) 18K3209 and (S) 15H05738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='M is supported by Grant-in-Aid for  Scientific Research (C) 20K03539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' Studies 131, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Press, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  [H1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Hille, On the volume of a tilting module, Abh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Hamburg 76 (2006), 261–277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' Hille, Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers, Lie algebras and  related topics, 91–101, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', 652, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', Providence, RI, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' Keller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Belg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' S´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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+page_content='  [NZ] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Nakanishi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Zelevinsky, On tropical dualities in cluster algebras, Algebraic groups and quantum groups,  217–226, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', 565, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=', Providence, RI, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  [Pl] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Plamondon, Generic bases for cluster algebras from the cluster category, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' IMRN 2013,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content=' 10, 2368–2420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='  30  TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO  Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita,  Osaka 565-0871, Japan  Email address: aoki-t@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='jp  Department of Pure and Applied Mathematics, Graduate School of Information Science and Tech-  nology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan  Email address: higashitani@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='jp  Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo  153-8914, Japan  Email address: iyama@ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='jp  Department of Information Science and Engineering, Okayama University of Science, 1-1 Ridaicho,  Kita-ku, Okayama 700-0005, Japan  Email address: r-kase@ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='jp  Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Met-  ropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan  Email address: yuya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
+page_content='mizuno@omu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'}
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