Coxeter elements and Kazhdan-Lusztig cells
type: Item
instance of: scholarly article
title: Coxeter elements and Kazhdan-Lusztig cells
zbMATH Open document ID: 1005.20032
author: Jian-yi Shi
published in: Journal of Algebra
publication date: 2002-12-03T00:00:00Z
review text: Let \((W,S,\Gamma)\) be a Coxeter system: a Coxeter group \(W\) with \(S\) the distinguished generator set and \(\Gamma\) the Coxeter graph. A Coxeter element of \(W\) is by definition a product of all generators \(s\in S\) in any fixed order. Let \(C(W)\) be the set of all the Coxeter elements in \(W\) and let \(C_0(W)=\bigcup_{J\subset S}C(W_J)\), where \(W_J\) is the standard parabolic subgroup of \(W\) generated by \(J\). For \(w\in C_0(W)\), denote by \(m(w)\) the maximal possible value of \(l(w_J)\) in an expression of the form \(w=x\cdot w_J\cdot y\). Denote by \(E(W)\) the set of all elements \(w\in C_0(W)\) such that \(m(sw)<m(w)\) for \(s\in{\mathcal L}(w)\), where \({\mathcal L}(w)=\{s\in S\mid l(sw)<l(w)\}\). In this paper the author gives an explicit expression for any distinguished involution \(d\) of \(W\) (\(W\) is of finite or affine type) with \(d\sim_Lw\) for some \(w\in E(W)\), i.e., if \(w=w_I\cdot y\) with \(m(w)=|I|\), then \(d=y^{-1}\cdot w_I\cdot y\), (for the notation \(d\sim_Lw\) and the definition of left cells see [\textit{D. Kazhdan} and \textit{G. Lusztig}, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]). The result verifies a conjecture proposed by the author [Conjecture 8.10, in Adv. Sci. China, Math. 3, 79-98 (1990)]. A subset \(K\) of \(W\) is left-connected, if for any \(x,y\in K\), there exists a sequence of elements \(x=x_0,x_1,\dots,x_r=y\) in \(K\) such that \(x_{i-1}x_i^{-1}\in S\) for \(1\leq i\leq r\). The author also shows that any left cell of \(W\) containing some element of \(C_0(W)\) is left-connected, which verifies a conjecture of \textit{G. Lusztig} [Adv. Stud. Pure Math. 6 255-287 (1985; Zbl 0569.20032)] in this case.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1762097
zbMATH Keywords: Coxeter elements
zbMATH Keywords: Kazhdan-Lusztig cells
zbMATH Keywords: Coxeter systems
zbMATH Keywords: Coxeter groups
zbMATH Keywords: generators
zbMATH Keywords: involutions
MaRDI profile type: Publication
full work available at URL: jabr.2001.9090
OpenAlex ID: W2048633473
cites work: Left cells in the weyl group of type7
cites work: Q4003858
cites work: Q5229495
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q3686887
cites work: Cells in affine Weyl groups. II
cites work: Q3767499
cites work: Q4206938
cites work: The Kazhdan-Lusztig cells in certain affine Weyl groups
cites work: A Two-Sided Cell in an Affine Weyl Group, II
cites work: The joint relations and the set \({\mathcal D}_ 1\) in certain crystallographic groups
cites work: Left cells in the affine Weyl group \(W_ a({\widetilde D}_ 4)\)
cites work: The enumeration of Coxeter elements
cites work: Left cells in the affine Weyl group of type \(\widetilde C_4\)
DOI: 10.1006/JABR.2001.9090
Recommended article: Q3322289
Recommended article: Note on a theorem of Eng
Recommended article: Some remarks on involutions in coxeter groups
Recommended article: Kazhdan-Lusztig cells in some weighted Coxeter groups
Recommended article: Centralizers of parabolic subgroups of Artin groups of type \(A_l\), \(B_l\), and \(D_l\)
Recommended article: The cells of the affine Weyl group \(\widetilde C_n\) in a certain quasi-split case. II.
Recommended article: 加权~Coxeter~群(C3,l6)的胞腔(英)
Recommended article: Some cells in the weighted Coxeter group (C̃n,ℓ̃2n)
Recommended article: ON ROOT SUBSYSTEMS AND INVOLUTIONS INSn
Recommended article: An Approach to the Connectedness of the Left Cells in Affine Weyl Groups
description: scientific article; zbMATH DE number 1762097