diff --git "a/B9AyT4oBgHgl3EQf4PrK/content/tmp_files/load_file.txt" "b/B9AyT4oBgHgl3EQf4PrK/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/B9AyT4oBgHgl3EQf4PrK/content/tmp_files/load_file.txt" @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf,len=1084 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='00784v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='RT] 2 Jan 2023 ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS LÉA BITTMANN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We interpret a formula established by Lapid-Mínguez on real regular rep- resentations of GLn over a local non-archimedean field as a matrix determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We use the Lewis Carroll determinant identity to prove new relations between real regular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Through quantum affine Schur-Weyl duality, these relations generalize Mukhin-Young’s Extended T -systems, for representations of the quantum affine algebra Uqppslkq, which are themselves generalizations of the celebrated T -system relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Preliminaries 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Good segments 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Determinant formula 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Extended T-system formula 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Relation to quantum affine algebras representations 17 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Ferrers boards 19 References 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Introduction The context of this work is the representation theory of GLnpFq (where F is a non- archimedean local field), or equivalently of the type A quantum affine algebra Uqppslkq (where q P Cˆ is not a root of unity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Indeed, through Chari-Pressley’s quantum affine Schur-Weyl duality [CP95], the category of complex smooth finite-length representations of GLnpFq is equivalent to the category of (level n) finite-dimensional Uqppslkq-modules, when k ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Since both contexts are equivalent, we will work with the category C of GLnpFq representations in most of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Both these categories have been intensively (and independently) studied, but some important natural questions remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The normalized parabolic induction, denoted by ˆ, endows this category with a ring category structure, and its Grothendieck group R with a ring structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Irreducible rep- resentations in the category C have been classified by Zelevinsky [Zel80] using multiseg- ments (formal sums of segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N a multisegment, the corresponding irreducible representation Zpmq is obtained as the unique irreducible sub- representation of the standard representation ζpmq :“ Zp∆1q ˆ Zp∆2q ˆ ¨ ¨ ¨ ˆ Zp∆Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The classes of the irreducible representations and the standard representations form two bases of the Grothendieck ring R, the change of basis matrix between them is unitriangular, with coefficients which can be expressed in terms of Kazhdan-Lusztig polynomials (see [Zel81], [CG97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A similar story was established for finite-dimensional representations of Uqppslkq (see the work of Nakajima [Nak01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' This gives an algorithm to compute the classes of 1 2 LÉA BITTMANN the simple representations from the classes of the standard representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' However, in practice the actual computation of the coefficients can be very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For some specific classes of irreducible representations, remarkable formulas have been established to compute their classes as linear combinations of classes of standard represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The work of Tadić [Tad95], and then Chenevier-Renard [CR08], established such a formula for Speh representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Cleverly, this formula can be seen as the computation of the determinant of a matrix, and it was then proved using the Lewis Carroll identity (also called Dodgson’s rule of determinant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In [LM14], Lapid-Mínguez generalized Tadić’s formula to a larger class of representations called ladder representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, in [LM18] the same authors established an even more general formula (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) below), for regular representations which are real - Zpmq such that Zpmq ˆ Zpmq is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Furthermore, in [LM14], Lapid-Mínguez used the Lewis Carroll identity to obtain a remarkable relation between the classes of some of these ladder representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For Zpmq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N) a ladder representation, we have the following relation in R [LM14, Corollary 12]: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq “ Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q ` Zpm1q ˆ Zpm2q, where Zpm1q, Zpm2q are also ladders (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Through quantum affine Schur- Weyl duality, relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) has been established independently by Mukhin-Young in [MY12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1] for representations of the quantum affine algebra Uqppslkq, under the name Extended T-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The extended T-systems are generalizations of the famous T-system relations, which are sets of recurrence relations of crucial importance in the study of certain integrable systems (see review [KNS11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For representations of quantum affine algebras, the T- systems are relations in the Grothendieck ring R between classes of Speh representations (called Kirillov-Reshetikhin modules there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' These relations were proved in all simply- laced types (A, D or E) by Nakajima [Nak01] and in all types by Hernandez [Her06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Additionally, the T-systems, and their extended version, can be interpreted as short exact sequences between irreducible finite-dimensional Uqppgq-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' More recently, the T-systems gained a new interpretation as exchange relations in a Fomin-Zelevinsky cluster algebra [FZ02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Indeed, in [HL16] Hernandez-Leclerc proved this interpretation of T-systems as cluster transformations and used it to the prove that the Grothendieck ring of the category of finite-dimensional Uqppgq-modules (in all Dynkin types) had the structure of a cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that Duan-Li-Luo obtained in [DLL19] another generalization of the T-systems, different from Mukhin-Young extended T-systems, which they also interpreted as exchange relations in the cluster algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In the present work, we establish formulas generalizing the extended T-systems of Mukhin-Young, for some real regular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Regular representations have a per- mutation associated to them and in [LM18], Lapid-Mínguez gave a sufficient condition for a regular representation to be real, as a pattern avoidance condition on the permutation associated to the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We show, using the notion of Ferres boards and the work of Sjostrand [Sjo07] that under the same pattern avoidance condition, Lapid-Mínguez’s formula [LM18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2 (9)] can be written as a matrix determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Our relations are then obtained using some choice of Lewis Carroll identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As our main result, we prove the following (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3): for Zpmq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N) a regular representation such that the associated permutation σ avoids the patterns 3412 ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 3 and 4231, we have the following relations in R ((5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2)): Zpmz∆Nq ˆ Zpmz∆σpNqq “ Zpmq ˆ Zpmz∆N, ∆σpNqq ` Zpm1 1q ˆ Zpm2 1q, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) Zpmz∆1q ˆ Zpmz∆σp1qq “ Zpmq ˆ Zpmz∆1, ∆σp1qq ` Zpm1 2q ˆ Zpm2 2q, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) where m1 1, m2 1, m1 2 and m2 2 are real regular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3, we also prove that, as the extended T-systems, these relations correspond to a decomposition of a module of length 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' the two terms in the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) are irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We prove this using Lapid-Mínguez’s [LM16] combinatorial irreducibility criteria, as well as a newly introduced notion of good segments in a mutlisegment, which enables us to prove by induction that some parabolic induction of irreducible representations are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We start with some reminders about segments, multi- segments, p-adic representations of GLnpFq and the Zelevinsky classification in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We also recall Lapid-Mínguez’s [LM16] irreducibility criteria for a parabolic induction of two representations, using socles and cosocles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In Section 3, we introduce the notion of good segments and use it to obtain some combinatorial criteria to prove that certain para- bolic inductions Zp∆q ˆ Zpmq, where Zpmq is a regular representation are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We also prove an existence result for good segments (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In particular, we obtain that every regular representation whose permutation avoids the patterns 3412 and 4231 has at least two good segments, from which we can recover that such representations are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In Section 4, we use the notion of Ferres boards and results from Sjostrand [Sjo07] and Chepuri–Sherman-Bennett [CSB21] to write existing relations as determinants of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The main result is stated and then proved in Section 5, in which we also give examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Finally, in Section 6 we translate our results to the context of quantum affine algebra representations, and give some perspective, in particular in relation to cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We would like to thank Alberto Mínguez for providing inspiration for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The author was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 948885 and by the Royal Society University Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Segments and multisegments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A segment is a pair of integers a ď b P Z, denoted by ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let Seg denote the set of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The extremities of the segment ∆ “ ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs P Seg are denoted by bp∆q “ a and ep∆q “ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We also write ÐÝ ∆ “ ra ´ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Two segments ∆ “ ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs and ∆1 “ rc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ds are linked if a ă c and c ´ 1 ď b ă d, or c ă a and a ´ 1 ď d ă b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In the first case, we say that ∆ precedes ∆1 and write ∆ ă ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A few examples of linked and unlinked pairs of segments: 1 3 4 5 are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1 2 4 5 are not linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1 4 3 5 are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1 5 2 4 are not linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4 LÉA BITTMANN Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A multisegment m is a finite formal sum of segments of Seg (with possible multiplicities), m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let Mult denote the set of multisegments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A sequence of segments p∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆Nq is said to be ordered if, for all 1 ď i ă j ď N, ∆i does not precedes ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If m P Mult, and p∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆Nq is an ordered sequence of segments such that m “ ∆1 ` ¨ ¨ ¨ ` ∆N, we say that p∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆Nq is an ordered form of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let F be a non-archimedean local field with a normalized absolute value | ¨ | and let D be a finite-dimensional central division F-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For n P Zě1, let CpGLnq be the category of complex, smooth representations of GLnpDq of finite length and IrrpGLnq the set of equivalence classes of irreducible objects of CpGLnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For πi P CpGLniq, i “ 1, 2, denote by π1ˆπ2 P CpGLn1`n2q the representation which is parabolically induced from π1 b π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The parabolic induction endows the category À Ně0 CpGLnq with the structure of a tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For any supercuspidal representation ρ P Ť nPZě0 IrrpGLnq, there exists a unique positive real number sρ such that ρ|¨|sρˆρ is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let νρ “ |¨|sρ, we write ÝÑρ “ ρνρ, ÐÝρ “ ρν´1 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A cuspidal line is an equivalence class on Ť nPZě0 IrrpGLnq for the equivalence relation given by ρ „ ÝÑρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For a fixed cuspidal line L, consider CL the Serre ring subcategory of À Ně0 CpGLnq consisting of the representations whose supercuspidal support is contained in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then all categories CL are equivalent as ring categories and the study of À Ně0 CpGLnq amounts to the study of one CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' From now on, we fix a cuspidal line, drop the subscript and consider the category C, its set of equivalence classes of irreducible objects Irr and its Grothendieck ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For ∆ “ ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs P Seg, consider the induced representation Ira;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs :“ ρνa ρ ˆ ρνa`1 ρ ˆ ¨ ¨ ¨ ˆ ρνb ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We consider the socle and cosocle of this representation: Zra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs :“ socpIra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bsq, maximal semi-simple submodule, Lra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs :“ cospIra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bsq, maximal semi-simple quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following is known (see for example [Zel80]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For ∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆N P Seg, Zp∆1qˆ¨ ¨ ¨ˆZp∆Nq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lp∆1qˆ¨ ¨ ¨ˆLp∆Nq) is irreducible if and only if the segments ∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆N are pairwise unlinked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For m P Mult and p∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ∆Nq an ordered form of m, define the standard module: ζpmq :“ Zp∆1q ˆ ¨ ¨ ¨ ˆ Zp∆Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' From the previous proposition, ζpmq does not depend on the chosen order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [Zel80][Zelevinsky Classification] The map m ÞÑ Zpmq :“ socpζpmqq, defines a bijection Mult „ ÝÑ Irr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Families of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We are interested in some particular families of rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let Zpmq be an irreducible representation, with m “ ∆1 ` ¨ ¨ ¨ ` ∆N P Mult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The irreducible representation Zpmq is a Speh representation if ∆i`1 “ ÐÝ ∆i, for all 1 ď i ď N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 5 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The representations corresponding to the multisegments r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 0s “ ‚0 ‚1 ‚2 ‚3 and r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s “ 4 2 1 0 3 2 are Speh representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The irreducible representation Zpmq is a ladder representation if, for all 1 ď i ď N ´ 1, ep∆i`1q ă ep∆iq and bp∆i`1q ă bp∆iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' All Speh representations are particular cases of ladder representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The representations corresponding to the multisegments r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 0s “ ‚0 1 2 5 3 , r4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 7s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s “ 4 7 2 1 are ladder representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The irreducible representation Zpmq is a regular representation if, for all 1 ď i ‰ j ď N, ep∆jq ‰ ep∆iq and bp∆jq ‰ bp∆iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' By extension, the multisegment m is also called regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' All ladder representations are particular cases of regular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The representation Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq “ 1 5 0 4 2 3 is a regular representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If Zpmq is a regular representation, then one can define a corresponding permutation σm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Write m “ ra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b1s ` ra2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b2s ` ¨ ¨ ¨ ` raN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNs, and assume b1 ą b2 ą ¨ ¨ ¨ ą bN, then σm P SN is such that aσmp1q ă aσmp2q ă ¨ ¨ ¨ ă aσmpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If Zpmq is a ladder representation, then the associated permutation is w0, the longest element of SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' An irreducible representation π is said to be real if π ˆ π is also irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Real representations are usually called square-irreducible representations in this context, but we use real here, which is the terminology coming from the work of Kang- Kashiwara-Kim-Oh [KKKO15] on representations of quantum affine algebras, where the notion appeared in a crucial way (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following is one of the main results of [LM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The regular representation Zpmq is real if and only if there does not exists a sequence 1 ď j1 ă ¨ ¨ ¨ ă jr ď N, r ě 4 such that if a1 i “ aji and b1 i “ bji, then either a1 i`1 ă a1 i ď b1 i`1 ` 1, i “ 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , r ´ 1, a1 3 ă a1 1 ď b1 3 ` 1, and a1 r ă a1 2 ă a1 r´1, or a1 i`1 ă a1 i ď b1 i`1 ` 1, i “ 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , r ´ 1, a1 4 ă a1 2 ď b1 4 ` 1, and a1 3 ă a1 r ă a1 1 ă a1 ℓ, 6 LÉA BITTMANN where ℓ “ 2 if r “ 4 and ℓ “ r ´ 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If the permutation σm avoids the patterns 4231 and 3412, then the condition of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='18 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We will call these representations pattern avoiding regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The same patterns avoidance condition correspond to the smoothness condition of the Schubert variety Xσm (see [LS90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that in particular, all ladder representations are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Irreducibility criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following result will be much used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [MS14] Let π1 and π2 be irreducible representations, and π be a represen- tation such that (a) π is a subrepresentation of π1 ˆ π2, (b) π is a quotient of π2 ˆ π1, (c) π1 b π2 has multiplicity 1 in the Jordan-Hölder sequence of π1 ˆ π2, Then π is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Given π1 “ Zpm1q and π2 “ Zpm2q, we write LIpπ1, π2q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' RIpπ1, π2q) for the condition Zpm1 ` m2q “ socpπ1 ˆ π2q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Zpm1 ` m2q “ cospπ1 ˆ π2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let m be a multisegment and ∆ a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then we have the following equivalences: LIpZp∆q, Zpmqq ðñ Zpm ` ∆q ãÑ Zp∆q ˆ Zpmq, RIpZp∆q, Zpmqq ðñ Zpm ` ∆q ãÑ Zpmq ˆ Zp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The first statement follows from the fact that the segment representation Zp∆q is a left multiplier (see [LM16, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3]), thus Zp∆q ˆ Zpmq has a unique irreducible submodule, which appears with multiplicity 1 in the Jordan-Hölder sequence of Zp∆q ˆ Zpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The second statement can be deduced from the first by the use of the contragredient, or more precisely [LM16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [LM16] π1 ˆ π2 is irreducible if and only if LIpπ1, π2q and RIpπ1, π2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In [LM16], Lapid-Minguez introduced a combinatorial setup in order to determine whether the conditions RIpZp∆q, Zpmqq and LIpZp∆q, Zpmqq where satisfied, for ∆ P Seg and m P Mult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us recall it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Write m “ ∆1 ` ¨ ¨ ¨ ` ∆N, and consider the sets X∆,m “ ti | ∆ ă ∆iu , ˜X∆,m “ ti | ∆i ă ∆u , Y∆,m “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' i | ÐÝ ∆ ă ∆i ) , ˜Y∆,m “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' i | ÐÝ ∆i ă ∆ ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let LCp∆, mq be the condition that there exists an injective function f : X∆,m Ñ Y∆,m such that for all 1 ď i ď N, ∆fpiq ă ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let RCp∆, mq be the condition that there exists an injective function f : ˜X∆,m Ñ ˜Y∆,m such that for all 1 ď i ď N, ∆i ă ∆fpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The function of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='23 are called matching functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [LM16] The conditions LCp∆, mq and LIpZp∆q, Zpmqq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' RCp∆, mq and RIpZp∆q, Zpmqq) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 7 Combining this result with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='22, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The parabolic induction Zp∆qˆZpmq is irreducible if and only if LCp∆, mq and RCp∆, mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Good segments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' A segment ∆ in a multisegment m P Mult is called a good segment if (i) Zp∆q ˆ Zpmq is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (ii) # Zpmq ãÑ Zp∆q ˆ Zpm´q, or Zpmq ãÑ Zpm´q ˆ Zp∆q, , where m´ “ mzt∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' second) subcase of (ii) is satisfied, ∆ is called a good left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' right) segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='21 as well as Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4, we have the following equivalences: ∆ is a good left segment of m ðñ LCp∆, mq, RCp∆, mq, and LCp∆, m´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) ∆ is a good right segment of m ðñ LCp∆, mq, RCp∆, mq, and RCp∆, m´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Combinatorial criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If m “ ∆1 ` ¨ ¨ ¨ ` ∆N is a multisegment and ∆0 is a segment such that ∆0 ` m is a regular multisegment, then LCp∆0, mq ô Ei, ∆0 ă ∆i, RCp∆0, mq ô Ei, ∆i ă ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We will prove the first equivalence, the second being exactly analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' From the definition of the condition LC, the implication LCp∆0, mq ð Ei, ∆0 ă ∆i is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let i P Y∆0,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If i R X∆0,m, then either bp∆0q “ bp∆iq or ep∆0q “ ep∆iq, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus Y∆0,m Ă X∆,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Now, if X∆0,m ‰ H, then LCp∆0, mq can not be satisfied (by Hall’s marriage theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If m “ ∆1 ` ¨ ¨ ¨ ` ∆N is an ordered regular multisegment with σ “ σm the associated permutation, then for all 1 ď i ď N, the condition LCp∆i, mq is equivalent to σ´1 is strictly decreasing on X∆i,m, the condition RCp∆i, mq is equivalent to σ´1 is strictly decreasing on ˜X∆i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As before, we only prove the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Fix 1 ď i ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If X∆i,m “ H, the equivalence is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Suppose X∆i,m ‰ H, then with the same reasoning as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2, Y∆i,m Ă X∆i,m Y tiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Suppose Y∆i,m “ X∆i,m Y tiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let X∆i,m “ tj1 ą j2 ą ¨ ¨ ¨ ą jmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, since m is ordered, ep∆j1q ă ep∆j2q ă ¨ ¨ ¨ ă ep∆jmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 8 LÉA BITTMANN If σ´1 is strictly decreasing on X∆i,m, then bp∆j1q ă bp∆j2q ă ¨ ¨ ¨ ă bp∆jmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Since all jk P X∆i,m, we have ∆jℓ ă ∆jℓ`1 for all 1 ď ℓ ď m ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus the function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) f : X∆,m Ñ Y∆,m, j1 ÞÑ i, jℓ`1 ÞÑ jℓ, 1 ď ℓ ď m ´ 1, is a matching function from X∆i,m to Y∆i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus LCp∆i, mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If Y∆i,m Ĺ X∆i,m Y tiu, then there exists j P X∆i,m such that bp∆jq “ ep∆iq ` 1, and Y∆i,m “ pX∆i,mztjuq Y tiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If σ´1 is strictly decreasing on X∆,m, then necessarily j “ jm and the function f from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) is a matching function from X∆i,m to Y∆i,m, as jm does not appear in the image of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Conversely, suppose LCp∆i, mq and let f be a matching function from X∆i,m to Y∆i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Necessarily, fpj1q “ i, as ∆i is the only segment considered which precedes ∆j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Recur- sively, we see that f is the function from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As it is a matching function, we deduce that ∆jℓ ă ∆jℓ`1 for all 1 ď ℓ ď m ´ 1, and thus σ´1 is strictly decreasing on X∆i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2, if m is a regular multisegment, for all 1 ď i ď k, LCp∆i, m´ ∆iq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' RCp∆i, m ´ ∆iq) is equivalent to the fact that ∆i precedes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' is preceded by) no segment in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Combining with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3, we have the following equivalences: ∆ is a good left segment of m ðñ ∆ precedes no other segment of m and ∆ forms a ladder with the segments which pre- cede it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ∆ is a good right segment of m ðñ ∆ is preceded by no other segment of m and ∆ forms a ladder with the segments which are preceded by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following result is clear using this criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If m1 is a sub-multisegment of m and ∆ P m1 is a good segment for m, then it is a good segment for m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For example, any segment ∆ is a good segment for itself, but not necessarily a good segment for any multisegment containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Existence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For N ě 2, let m “ ∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N be a regular multisegment such that for all i, ∆i “ rai, bis with b1 ą b2 ą ¨ ¨ ¨ ą bN and the associated permutation σ avoids the patterns 4231 and 3412, and π “ Zpmq is a prime irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then either ∆1 or ∆σp1q, and either ∆N or ∆σpNq correspond to good segments of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, if σpNq “ 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' σp1q “ N), then ∆N is a good right segment (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ∆1 is a good left segment) of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If σpNq “ 1 and σp1q “ N then m is a ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' First of all, if σp1q “ 1, then ∆1 is not linked with any other segment of m, and it is both a good left and a good right segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let i0 “ σp1q and suppose i0 ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Suppose neither ∆1 nor ∆i0 are good segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3, σ´1 m is neither decreasing on X∆1,m nor on ˜X∆i0,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We consider different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If there exists i ă j ă i0 such that ∆i ă ∆1 and ∆j ă ∆1, or ∆i0 ă ∆i and ∆i0 ă ∆j and ∆j ć ∆j, the configuration is the following: ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 9 ∆i0 ∆j ∆i ∆1 The pattern 4231 appears in this configura- tion, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Otherwise, there exists at least one 1 ă i ă i0 such that ∆i0 ă ∆i and ∆i ć ∆1 and one i0 ă j such that ∆j ă ∆1 and ∆j ć ∆i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The configuration is the following: ∆j ∆i0 ∆i ∆1 The pattern 3412 appears in this configura- tion, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The proof for ∆N and ∆σpNq is exactly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Now, suppose σpNq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We know that either ∆1 or ∆N is a good segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If ∆1 is a good segment and ∆N is not a good segment, then we are necessarily in the first configuration drawn above, which features the pattern 4231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, if σp1q “ N, then either ∆1 or ∆N is a good segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The same pattern avoidance condition implies that ∆1 is necessarily good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If both σpNq “ 1 and σp1q “ N then any pair of segments between ∆1 and ∆N which does not form a ladder would create a 4231 pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus m is a ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' □ This result has the following consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let π “ Zpmq be a regular representation avoiding the patterns 4231 and 3412, then π has at least two good segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' This criteria allows us to recover the implication (which is established by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='18 [LM18]): m avoids the patterns 4231 and 3412 ùñ Zpmq is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' This can be proved by induction on N, the number of segments in the multisegment m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For completeness, let us detail the reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If N “ 1, then m “ ∆ is just a segment, and Zp∆q is real (for example as an application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If N ě 2 and m avoids the patterns 4231 and 3412, then from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8, m has at least one good segment ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Suppose without loss of generality that it is a good left segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then Zpmq ˆ Zpmq ãÑ Zpmq ˆ Zp∆q looooooomooooooon irreducible ˆZpm´q, ãÑ Zp∆q ˆ Zpmq ˆ Zpm´q ãÑ Zp∆q ˆ Zp∆q looooooomooooooon irreducible ˆZpm´q ˆ Zpm´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' However, m´ has N ´ 1 segments, and satisfy the pattern avoidance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus Zpm´q ˆ Zpm´q is irreducible by induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, as Zpmq և Zpm´q ˆ Zp∆q, Zpmq ˆ Zpmq և Zpm´q ˆ Zpm´q ˆ Zp∆q ˆ Zp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then the irreducibility of Zpmq ˆ Zpmq is obtained through Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Notice we only used the existence of one good segment in the proof, although there is two from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 10 LÉA BITTMANN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Determinant formula 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Alternate sum formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' One of the results of [LM18] is an alternate sum formula for every regular real representation using standard representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let π “ Zpmq be a regular real representation, with m “ ra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b1s ` ¨ ¨ ¨ ` raN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNs such that b1 ą ¨ ¨ ¨ ą bN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In the Grothendieck ring, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) π “ ÿ σ1PSN σ0ďσ1ďσ sgnpσ1σqZpraσp1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p1qsq ˆ Zpraσp2q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p2qsq ˆ ¨ ¨ ¨ ˆ ZpraσpNq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1pNqsq, where σ “ σm and for all i, σ´1 0 piq “ max ␣ j ď xi | j R σ´1 0 pti ` 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , Nuq ( , with xi “ #tj | aj ď bi ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The permutation σ0 satisfies σ0 ď σ1 ô @i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , Nu, aσpiq ď bσ1piq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We deduce that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) is equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) π “ ÿ σ1PSN σ1ďσ sgnpσ1σqZpraσp1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p1qsq ˆ Zpraσp2q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p2qsq ˆ ¨ ¨ ¨ ˆ ZpraσpNq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1pNqsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Indeed, for σ1 ą σ0, at least one of the Zpraσpiq, bσ1piqsq is not defined, and the term does not contribute to the sum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For all i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , Nu, set a1 i “ aσpiq, then equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) can be rewritten (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) π “ sgnpσq ÿ σ1PrId,σs sgnpσ1qZpra1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p1qsq ˆ Zpra1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1p2qsq ˆ ¨ ¨ ¨ ˆ Zpra1 N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσ1pNqsq, where rId, σs denotes the Bruhat interval of permutations in SN lower than σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Matrix determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) is similar to the determinant of a matrix, with some entries replaced by zeros to account for the missing permutations σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' More precisely, for σ1, σ2 permutations in SN, let Γrσ1, σ2s :“ tpi, σpiqq | σ P rσ1, σ2s, 1 ď i ď Nu, then permutations whose graph is contained in ΓrId, σs form the right convex hull, from the work of Sjöstrand [Sjo07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following is obtained using [Sjo07, Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [CSB21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3] If the permutation σ P SN avoids the patterns 4231 and 34121, and M “ pmi,jq1ďi,jďN is a square N ˆ N-matrix, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4) detpM|ΓrId,σsq “ ÿ σ1PrId,σs sgnpσ1qm1,σ1p1qm2,σ1p2q ¨ ¨ ¨ mN,σ1pNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Using Ferrers boards (see Appendix A), the determinant in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4) can be computed placing the coefficient mi,j in the box pi, jq of rNs2 if it is coloured and 0 if it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that the dots are placed on the Zprai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bisq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Combining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2 with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3), and assuming σ avoids the patterns 4231 and 3412, we obtain the following: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5) π “ sgnpσq det ` pZpa1 i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjqq1ďi,jďN|ΓrId,σs ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1in [Sjo07, CSB21], the pattern avoidance condition is weaker, permutations are assumed to avoid the patterns 4231, 35142, 42513, and 351624 ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lewis Carroll’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following result is usually called the Lewis Carroll’s identity or the Desnanot–Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For M a square N ˆ N-matrix, and A, B Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , Nu, let MB A be the matrix obtained from M by removing all rows indexed by elements of A and all columns indexed by elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, for all 1 ď a ă a1 ď N and 1 ď b ă b1 ď N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6) detpMq detpMb,b1 a,a1q “ detpMb aq detpMb1 a1q ´ detpMb1 a q detpMb a1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We can use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4 with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5) to write relations in the Grothendieck ring R involving π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' However, if M “ ` pZpa1 i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjqq1ďi,jďN|ΓrId,σs ˘ , the determinant of the submatrix Mj i does not necessarily realize (up to a sign) the class of an irreducible repre- sentation Zpm1q in R, for all 1 ď i, j ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let m “ r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s`r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s`r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s, the corresponding permutation is the reflection σ “ p12q P S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The alternate sum formula for the class of the irreducible representation is Zpmq “ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq ˆ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ˆ Zpr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq ´ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ˆ Zpr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq, “ ´ ������ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq 0 Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq 0 0 0 Zpr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let M be the above matrix, then detpM1 2 q “ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ˆ Zpr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq “ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq P R, detpM1 3 q “ 0 ‰ Zpm1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Nevertheless, it is possible to write explicit formulas in the Grothendieck ring R in some interesting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We will use the following key result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' [CSB21, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='17] Let σ be a permutation in SN avoiding the patterns 4231 and 3412, and choose i P rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let σi P SN´1 be the "flatten" permutation obtained from σ by removing pi, σpiqq and shifting the remaining numbers appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then for M a pN ´ 1q ˆ pN ´ 1q-matrix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7) detpM|ΓrId,σsσpiq i q “ detpM|ΓrId,σisq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note for M a N ˆ N-matrix and for 1 ď i, j ď N, ` M|ΓrId,σs ˘j i “ Mj i |ΓrId,σsj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Extended T-system formula Our main result is the following, which will be proven in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let m “ ∆1`∆2`¨ ¨ ¨`∆N be a regular multisegment, such that b1 ą b2 ą ¨ ¨ ¨ ą bN, where for all 1 ď i ď N, ∆i “ rai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Assume the corresponding permutation σ avoids the patterns 4231 and 3412, and that σpNq ‰ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let I “ " i | aN ď ai bi ď bσpNq “ ti1 ă i2 ă ¨ ¨ ¨ iru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, we have the following relation, in the Grothendieck ring R: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) Zpmz∆Nq ˆ Zpmz∆σpNqq “ Zpmq ˆ Zpmz∆N, ∆σpNqq ` Zpm1q ˆ Zpm2q, 12 LÉA BITTMANN where m1 “ ÿ jRI ∆j ` r´1 ÿ k“1 raik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bik`1s, m2 “ ÿ iRI ∆i ` r´1 ÿ k“1 raik`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' biks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, the products in both terms on the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (1) If σpNq “ N, then the segment ∆N is not linked to any other segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In that case Zpmq “ Zpmz∆Nq ˆ Zp∆Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (2) As σ avoids the pattern 4231, the segments ∆i, with i P I form a ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us assume the permutation σ avoids the patterns 4231 and 3412 and satisfies σp1q ‰ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let J “ " j | aj ď a1 bσp1q ď bj “ tj1 ă j2 ă ¨ ¨ ¨ jsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The following relation in satisfied in the Grothendieck ring R: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) Zpmz∆1q ˆ Zpmz∆σp1qq “ Zpmq ˆ Zpmz∆1, ∆σp1qq ` Zpm1q ˆ Zpm2q, where m1 “ ÿ iRJ ∆i ` s´1 ÿ k“1 rajk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjk`1s, m2 “ ÿ iRJ ∆i ` s´1 ÿ k“1 rajk`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The result is obtained by applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 to the irreducible representation Zpm1q, with m1 “ r´bN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ´aNs ` ¨ ¨ ¨ ` r´b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ´a1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (1) Let m “ r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s`r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s`r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The corresponding regular represen- tation Zpmq is real, since its associated permutation is σ “ 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' It has two good right segments, which are r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s and r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1s (∆σp1q and ∆3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 gives the following relation: Zpr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq ˆ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1sq “ Zpmq ˆ Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq ` Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2s`r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1sq – Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sqˆZpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 1sq, and in this case the above relation can be simplified by Zpr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 2sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (2) Let m “ r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s`r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s`r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s`r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The corresponding regular representation Zpmq is real, since its associated permutation is σ “ 3142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' It has two good segments, which are r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s (left) and r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 gives the following relation: Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq “ Zpmq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq ` Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Whereas applying Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3 gives the following relation: Zpr3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ˆ Zpr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6s ` r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq “ Zpmq ˆ Zpr3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3sq ` Zpr3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 4sq ˆ Zpr3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5s ` r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 3s ` r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Ladder case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If m is a ladder, then the corresponding permutation is the longest permutation w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus ΓrId, w0s “ rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In that case, the result of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 is al- ready known, as Corollary 12 of [LM14], or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 in [MY12], in the language of representations of quantum affine algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let m “ ∆1 ` ¨ ¨ ¨ ` ∆N be a ladder multisegment, with ∆i “ rai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bis, then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3) Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq “ Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q ` Zpra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b2s ` ¨ ¨ ¨ ` raN´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNsq ˆ Zpra2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b1s ` ¨ ¨ ¨ ` raN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bN´1sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In this case, the result comes from the application of the Lewis Carroll identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6) to the matrix pZprai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjsqq1ďi,jďN, on lines and columns 1 and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' However, in order to understand better the general case, let us consider in more details the application of the Lewis Carroll identity to the matrix M “ pZpra1 i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjsqq1ďi,jďN (recall that a1 i “ aN´i`1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' One can look at what happens to the Ferrers boards (see Appendix A) in this case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The permutation w0 is represented by an anti-diagonal, and the Ferrers board is the full grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Taking out row 1 and column N, one gets exactly the grid corresponding to the longest element of SN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' S5 Q p15qp24q “ ‚ ‚ ‚ ‚ ‚ ÝÑ ‚ ‚ ‚ ‚ “ p14qp23q P S4 As the signature of the longest permutation in SN is p´1qt N 2 u, one has p´1qt N´1 2 u detpM1 Nq “ Zp∆1 ` ¨ ¨ ¨ ` ∆N´1q, p´1qt N´1 2 u detpMN 1 q “ Zp∆2 ` ¨ ¨ ¨ ` ∆Nq, p´1qt N 2 u´1 detpM1,N 1,N q “ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Now, taking out row 1 and column 1, or row N and column N, one gets again the grid corresponding to the longest element of SN´1, but the dots have moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For example, if one does a cyclic permutation of the columns by shifting them to the left and placing column 1 at the end, then taking out row 1 and column 1 gives the same result as taking out row 1 and column N in the shifted board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ‚ ‚ ‚ ‚ ‚ ÝÑ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ÝÑ ‚ ‚ ‚ ‚ ‚ ‚ ‚ The same operation can be applied to the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that the new dots are placed on the coefficients Zpra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b2sq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , ZpraN´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The permutation of the columns does not change the sign of the determinant because the columns on which the determinant is computed are not permuted with respect to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' detpM1 1 q “ detpshiftpMqN 1 q “ p´1qt N´1 2 uZpra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b2s ` ¨ ¨ ¨ ` raN´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, detpMN N q “ p´1qt N´1 2 uZpra2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b1s ` ¨ ¨ ¨ ` raN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bN´1sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 14 LÉA BITTMANN Finally, the Lewis Carroll identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6) gives relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, the irreductibility of the terms Zpmq ˆ Zp∆2 ` ¨ ¨ ¨ ` ∆N´1q and Zpra1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b2s ` ¨ ¨ ¨ ` raN´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNsq ˆ Zpra2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' b1s ` ¨ ¨ ¨ ` raN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bN´1sq is proven in [BLM13, Exemple 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof of relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4 (Lewis Carroll’s identity) to the matrix ˜ M “ M|ΓrId,σs, where M “ ppZpa1 i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjqq1ďi,jďNq, on rows σ´1pNq, N and columns σpNq, N: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4) detp ˜ Mq detp ˜ MσpNq,N σ´1pNq,Nq “ detp ˜ MσpNq σ´1pNqq detp ˜ MN N q ´ detp ˜ MN σ´1pNqq detp ˜ MσpNq N q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='6, det ´ ˜ MσpNq N ¯ “ det ´ MσpNq N |ΓrId,σNs ¯ , det ´ ˜ MN σ´1pNq ¯ “ det ´ MN σ´1pNq|ΓrId,σσ´1pNqs ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Since σN and σσ´1pNq satisfy the pattern avoidance condition, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5), one has det ´ MσpNq N |ΓrId,σNs ¯ “ sgnpσNqZp∆1 ` ¨ ¨ ¨ ` { ∆σpNq ` ¨ ¨ ¨ ` ∆Nq, det ´ MN σ´1pNq|ΓrId,σσ´1pNqs ¯ “ sgnpσσ´1pNqqZp∆1 ` ∆2 ` ¨ ¨ ¨ ` ∆N´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that for i P rNs, Mσpiq i is obtained by taking out the row and column containing the coefficient Zpra1 i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσpiqsq “ Zp∆σpiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, det ´ ˜ MσpNq,N σ´1pNq,N ¯ “ sgnpσσ´1pNqN qZp∆1 ` ¨ ¨ ¨ ` { ∆σpNq ` ¨ ¨ ¨ ` ∆N´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us now consider the coefficients detp ˜ MN N q and detp ˜ MσpNq σ´1pNqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3, we know that either the two columns σpNq and N or the two rows σ´1pNq and N of ˜ M are similar (the zeros are at the same place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Assume that the two columns σpNq and N of ˜ M are similar, meaning that there are as many zeros above the coefficient Zpra1 σ´1pNq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bσpNqsq as above Zpra1 σ´1pNq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bNsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that since σ avoids the pattern 4231, the dots in the lower right corner of the Ferrers board form a ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In particular, one can apply the same reasoning as in the ladder case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us cyclically permute the columns σpNq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , N in order to obtain column N in position σpNq, and take the determinant of the minor shiftp ˜ MN N q “ shiftp ˜ MqσpNq N instead of the minor ˜ MN N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Because theses columns have the same zero block in their upper part, these determinants are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The resulting permutation is σN (same permutation as if we had taken out row N and column σpNq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ‚ ‚ ‚ ‚ ‚ ÝÑ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ÝÑ ‚ ‚ ‚ ‚ ‚ For j R I, there are still dots placed on the Zpraj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bjsq, and the new dots are placed on the Zpraik`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' biksq, for 1 ď k ď r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Hence, detp ˜ MN N q “ detpshiftp ˜ MqσpNq N q “ sgnpσNqZpm2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 15 Similarly, detp ˜ MσpNq σ´1pNqq “ sgnpσσ´1pNqqZpm1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us now consider the signatures of the permutations, using the criteria of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4: sgnpσq “ ÿ ‚ 7 " ‚ ‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us separate the grid (or matrix) in 4 blocks, A, B, C and the upper right being empty (or filled with zeros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' p0q A B C ‚ pN, σpNqq ‚ pσ´1pNq, Nq Note that zone C contains r dots, where r is the cardinal of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Going from σ to σN, we take out the dot on the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' It is clear that all dots in zones A, B and C except the bottom one will have the same contribution to the sum ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The difference is thus equal to the contribution of the bottom dot pN, σpNqq, which is r ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Hence sgnpσNq “ sgnpσqp´1qr´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Now, going from σ to σσ´1pNq, we take out the dot pσ´1pNq, Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' All dots is block A will still have the same contribution, but the dots in block B will count one less dot in their upper right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The remaining dots in block C will also count one less dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus, sgnpσσ´1pNqq “ sgnpσqp´1qr´1`7B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Going from σ to σσ´1pNqN , we take out both these dots, and the signature of the resulting permutation is sgnpσσ´1pNqN q “ sgnpσqp´1q7B`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Simplifying the signs in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4), we get the desired relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Finally, in the case where it is not the two columns but the two rows σ´1pNq and N of ΓrId, σs which are identical, one can apply the same procedure of cyclically permuting the rows of the matrix ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As a result, detp ˜ MN N q “ sgnpσσ´1pNqqZpm2q, detp ˜ MσpNq σ´1pNqq “ sgnpσNqZpm1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' But a symmetric reasoning on the signatures, we also get the desired relation, which concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof of irreducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Irreductibility of ZpmqˆZpmz∆N, ∆σpNqq: As in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='9, let us prove this result by induction on N ě 3, the number of segments in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For N “ 3, assuming σp3q ‰ 3, then Zpmz∆3, ∆σp3qq “ Zp∆q, where ∆ is necessarily a good segment of m by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' By definition, Zpmq ˆ Zp∆q is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let N ě 4, from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7, as σ avoids the patterns 4231 and 3412, we know that either m is a ladder or it has at least one good segment ∆ which is different from ∆N and ∆σpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The ladder case has been considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Otherwise, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5, we know that ∆ is also a good segment of m1 :“ mz∆N, ∆σpNq (on the same side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 16 LÉA BITTMANN We can assume without loss of generality that ∆ is a good left segment of m and m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, as in the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8, Zpmq ˆ Zpm1q ãÑ Zpmq ˆ Zp∆q looooooomooooooon irreducible ˆZpm1z∆q – Zp∆q ˆ Zpmq ˆ Zpm1z∆q ãÑ Zp∆q ˆ Zp∆q looooooomooooooon irreducible ˆZpmz∆q ˆ Zpm1z∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' By induction, Zpmz∆q ˆ Zpm1z∆q is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, Zpmq ˆ Zpm1q և Zpmz∆q ˆ Zpm1z∆q ˆ Zp∆q ˆ Zp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We conclude that Zpmq ˆ Zpm1q is irreducible by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Irreducibility of Zpm1q ˆ Zpm2q: As before, we prove this by induction, this time on N ´ r, where r “ |J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If r “ N, then m is a ladder, and the result was proven in [BLM13, Exemple 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If N ą r, then m is not a ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In that case, either ∆1 or ∆σp1q is a good segment of m and does not form a ladder with ∆N and ∆σpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' This good segment is not one of the ∆ik and thus it is a segment of m1 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us prove it is a common good segment of m1 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us assume that σpNq ‰ 1 and that ∆1 is a good left segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Clearly, ∆1 does not precede any segment of m1 or m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Suppose ∆1 does not form a ladder with the segments which precedes it in m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus there exists ∆, ∆1 such that ∆ ă ∆1, ∆1 ă ∆1 and ∆1 Ĺ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As ∆1 is a good segment of m, necessarily exactly one of ∆, ∆1 is in m while the other has be shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If ∆1 P m, then ∆ “ raik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bik`1s for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In that case, ∆ik`1 ă ∆1 and ∆1 ć ∆ik`1, which contradicts the fact that ∆1 is a good segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ∆1 bik`1 ∆1 aik aik`1 If ∆ “ rai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bis P m, then there is k such that ∆1 “ raik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bik`1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If both bik ą bi and aik`1 ă ai then i P I, which contradicts the fact that ∆ has not been shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ∆1 ∆ bik`1 aik aik`1 bik aik If bik ă bi, then ∆ik, ∆, ∆1 do not form a ladder in m, if aik`1 ą ai then ∆, ∆ik`1, ∆1 do not form a ladder in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In both cases, it contradicts the fact that ∆1 is a good segment of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ∆1 ∆ bik`1 aik bik aik ∆1 ∆ bik`1 aik aik`1 Hence by the criteria in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2, ∆1 is good segment of m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We show in a similar way that ∆1 is good segment of m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then, Zpm1q ˆ Zpm2q ãÑ Zp∆1q ˆ Zp∆1q loooooooomoooooooon irreducible ˆZpm1z∆1q ˆ Zpm2z∆1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 17 By induction, Zpm1z∆1q ˆ Zpm2z∆1q is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Similarly, Zpm1q ˆ Zpm2q և Zpm1z∆1q ˆ Zpm2z∆1q ˆ Zp∆1q ˆ Zp∆1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We conclude that Zpm1q ˆ Zpm2q is irreducible by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Relation to quantum affine algebras representations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Translation of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As mentioned above, the result of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 has a quan- tum affine analog through quantum affine Schur-Weyl duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Indeed, when q is not a root of unity, Chari-Pressley [CP96] have established an equivalence of categories between the category of finite-dimensional representations of the affine Hecke algebra 9Hq2pnq and the category of (level n) finite-dimensional representations of the quantum affine algebra Uqppslkq, when k ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, through type theory (see for example [Hei11]), it is known that finite-dimensional representations of the affine Hecke algebra 9Hq2pnq are equivalent to finite length representations of GLnpFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' This equivalence is monoidal, in the sense that the parabolic induction of two represen- tations in C is translated into the tensor product of the corresponding Uqppslkq-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Instead of multisegments, finite-dimensional irreducible Uqppslkq-modules have been clas- sified [CP95] using Drinfeld polynomials, which correspond to their highest-weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' By a process similar to the reduction to cuspidal lines described in the beginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2, the study of the category of finite-dimensional Uqppslkq-modules amounts to the study of a skeleton Serre subcategory C , introduced by Hernandez-Leclerc (see [HL10, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='7]), in relation to cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let R denote the Grothendieck ring of the monoidal cate- gory C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Simple objects in the category C are then parametrized, up to isomorphism, by mono- mials in the formal variables Yi,p, pi, pq P ˆI :“ tt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' , k ´ 1u ˆ Z | i ` p ` 1 P 2Zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The correspondence between segments and formal variables is as follows: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' bs ÞÑ Yb´a`1,´a´b, r1´i´p 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' i´p´1 2 s Ð� Yi,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Since we are using the Zelevinsky classification for the representations of GLnpFq, from now on irreducible Uqppslkq-modules will be denoted LpMq, with M their highest loop- weight, in the set of dominant loop-weights: ˆPℓ :“ # N ź j“1 Yij,pj | @ 1 ď j ď N, pij, pjq P ˆI + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Through this correspondence, ladder representations are usually called snake modules in the context of quantum affine algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For completeness, recall the definition of snakes modules by Mukhin-Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For M “ śN j“1 Yij,pj P ˆPℓ, the simple module LpMq is a snake module if and only if, for all 1 ď j ď N, pj`1 ´ pj ě |ij`1 ´ ij| ` 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' It clearly translates to the definition of ladders, as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that a definition of snake modules for type B quantum affine algebras was also introduced by Mukhin-Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, as stated above, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='5 from [LM14] was previously established by Mukhin-Young in terms of snake modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 18 LÉA BITTMANN For M “ śN j“1 Yij,pj P ˆPℓ such that LpMq is a snake module, we have the following relation, in the Grothendieck ring R [MY12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1]: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) « L ˜N´1 ź j“1 Yij,pj ¸ff ¨ « L ˜ N ź j“2 Yij,pj ¸ff “ « L ˜N´1 ź j“2 Yij,pj ¸ff ¨ rLpMqs ` “ LpM1q ‰ ¨ “ LpM2q ‰ , where M1, M2 are called the neighboring snakes of M, and correspond to m1 and m2 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) was established in [MY12] also in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, as in our result, both terms on the left hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) correspond to irreducible modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For these reasons, our theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 is a generalization of [MY12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1], and we have established some new relations between irreducible representations of Uqppslkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Let us translate the relations obtained in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4: (1) For k ě 4, let M “ Y2,´5Y3,´2Y1,´2 P ˆPℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' As before, the corresponding regular representation LpMq is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 gives the following relation: LpY2,´5Y3,´2q ¨ LpY3,´2Y1,´2q “ LpMq ¨ LpY3,´2q ` LpY3,´2q ˆ LpY3,´4Y3,´2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' (2) For k ě 7, let M “ Y6,´7Y3,´8Y5,´4Y2,´5 P ˆPℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The corresponding regular repre- sentation LpMq is real and applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 gives the following relation: LpY6,´7Y3,´8Y5,´4q ¨ LpY6,´7Y5,´4Y2,´5q “ LpMq ¨ LpY6,´7Y5,´4q ` LpY6,´7Y5,´4Y1,´6q ¨ LpY6,´7Y5,´4Y4,´7q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Whereas applying Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3 gives the following relation: LpY3,´8Y5,´4Y2,´5q ¨ LpY6,´7Y3,´8Y2,´5q “ LpMq ¨ LpY3,´8Y2,´5q ` LpY3,´8Y2,´5Y4,´5q ¨ LpY3,´8Y2,´5Y7,´6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Note that when k “ 7, the right hand side of the last relation simplifies as LpY3,´8Y2,´5Y4,´5q ¨ LpY3,´8Y2,´5q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Relation to cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In [HL16], Hernandez and Leclerc proved that the Grothendieck ring R had a cluster algebra structure for which the initial cluster variables are Kirillov-Reshetikhin modules (or Speh representations, as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Moreover, one of the key ingredients used for this result is the fact that the T-system relations (of which the Mukhin-Young extended T-systems are generalizations) correspond to exchange relations in the cluster algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The same authors also conjectured [HL16, Con- jecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2] that the cluster variables were in bijection with the prime real simple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Part of this conjecture was proven by Kashiwara-Kim-Oh-Park in [KKOP21], where they proved that all cluster variables correspond to prime real simple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' In [DLL19] Duan-Li-Luo proved that prime snake modules correspond to cluster vari- ables, thus proving Hernandez-Leclerc’s conjecture for snake modules, and for that purpose introduced new relations in the Grothendieck ring R, which they interpreted as exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' However, it is unclear whether (some of) the Mukhin-Young extended T-systems can be interpreted as exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' One of the motivations behind this work was to obtain more generalizations of the T- system relations, which could be interpreted as exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' We conjecture that, equipped with more explicit relations such as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2), one could prove that all prime real regular representations (for which there exists the criterion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='18 [LM18]) correspond to cluster variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ON A DETERMINANT FORMULA FOR SOME REAL REGULAR REPRESENTATIONS 19 However, we already observe that not all relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2) have the form of an exchange relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For example, in the relation in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1 (1), one of the factors in the left hand side is not prime LpY3,´2Y1,´2q – LpY3,´2q¨LpY1,´2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus the left hand side is a product of three prime irreducible modules, and the relation cannot be an exchange relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Ferrers boards Permutations in SN can be represented by placing dots in an N ˆ N-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' For all 1 ď i ď N, place a dot in the box pi, σpiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Then the set ΓrId, σs Ă rNs2 can be represented in the grid by colouring the boxes pi, σ1piqq, for σ1 ď σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The grid corresponding to the permutation σ “ 152463 is ‚ ‚ ‚ ‚ ‚ ‚ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' By the study of Sjöstrand [Sjo07], the set ΓrId, σs is a right-aligned Skew Ferrers board, in particular it is the union of the rectangles ‚ ‚ for all pairs of (not necessarily distinct) dots ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The σ be a permutation in SN which avoids the pattern 3412, then either the columns σpNq and N or the rows σ´1pNq and N of ΓrId, σs are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' If the columns σpNq and N of ΓrId, σs are different, then there is a dot above pσ´1pNq, Nq and to the right of pN, σpNqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ‚ ‚ ‚ pσ´1pNq, Nq pN, σpNqq Similarly, if the rows σ´1pNq and N are different, then there is a dot to the left of pN, σpNqq and below pσ´1pNq, Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Thus if both the columns σpNq and N and the rows σ´1pNq and N are different, then there a 3412 configuration, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' ‚ ‚ ‚ ‚ □ The following is clear from the definition of the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' The signature of the permutation σ is equal to p´1qℓ, where ℓ is the sum over all dots ‚ of the number of dots strictly above and to the right of ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQf4PrK/content/2301.00784v1.pdf'} +page_content=' 20 LÉA BITTMANN References [BLM13] I.' metadata={'source': 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