Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials.
type: Item
instance of: scholarly article
title: Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials.
zbMATH Open document ID: 1047.20038
DOI: 10.1016/S0001-8708(02)00100-7
author: Fokko du Cloux
published in: Advances in Mathematics
publication date: 2004-01-06T00:00:00Z
review text: Let \(W=(W,S)\) be a Coxeter group with \(S\) its distinguished generator set. Denote by \(\leqslant\) the Bruhat ordering on \(W\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined polynomials \(P_{x,y}\in\mathbb{Z}[q]\) for each \(x\leqslant y\) in \(W\) [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. These polynomials play an important role in representation theory. \textit{M. Dyer} conjectured [Compos. Math. 78, No. 2, 185-191 (1991; Zbl 0784.20019)] that \(P_{x,y}\) depends only on the isomorphism type of the poset \([x,y]=\{z\in W\mid x\leqslant z\leqslant y\}\) for the Bruhat ordering. The main result of the present paper is to give an affirmative answer to the conjecture for a class of groups, which can be stated as follows. Let \(W,W'\) be two Coxeter groups. Let \(w\in W\) and \(w'\in W'\) be such that the posets \([e,w]\) and \([e',w']\) are isomorphic for the Bruhat orderings on \(W,W'\), where \(e,e'\) are the identity elements of \(W,W'\), respectively. Then any poset isomorphism \(\phi\colon[e,w]\to[e',w']\) preserves Kazhdan-Lusztig polynomials, (in the sense that \(P_{\phi(x),\phi(y)}=P_{x,y}\) for any \(x\leqslant y\) in \([e,w]\)) in the case where one of the two groups \(W,W'\) has the property that the Coxeter graph of each of its irreducible constituents is either a tree or affine of type \(\widetilde A_n\). In particular, the result holds for all finite or affine Coxeter groups. Note that the above result was also obtained by \textit{F. Brenti} [The intersection cohomology of Schubert varieties is a combinatorial invariant, preprint (2002)] in the case where \(W,W'\) are both of type \(A_n\), as a corollary of a purely combinatorial construction of the \(P_{x,y}\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 2022086
zbMATH Keywords: Bruhat orderings
zbMATH Keywords: Kazhdan-Lusztig polynomials
zbMATH Keywords: Schubert varieties
zbMATH Keywords: Coxeter groups
describes a project that uses: Coxeter
MaRDI profile type: Publication
cites work: Q5580319
cites work: A combinatorial formula for Kazhdan-Lusztig polynomials
cites work: The intersection cohomology of Schubert varieties is a combinatorial invariant
cites work: Computing Kazhdan-Lusztig Polynomials for Arbitrary Coxeter Groups
cites work: An abstract model for Bruhat intervals
cites work: Q3977655
cites work: Q4865649
cites work: Q4776063
cites work: Q3998721
cites work: Representations of Coxeter groups and Hecke algebras
cites work: The enumeration of Coxeter elements
cites work: Q4344108
cites work: Automorphisms of the Bruhat order on Coxeter Groups
Recommended article: Q3977655
Recommended article: Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity.
Recommended article: Q4735310
Recommended article: Q4338604
Recommended article: Special matchings and Kazhdan-Lusztig polynomials
Recommended article: On the first coefficients in \(q\) of the Kazhdan-Lusztig polynomials
Recommended article: Leading coefficients of Kazhdan-Lusztig polynomials and fully commutative elements.
Recommended article: Boolean elements in Kazhdan-Lusztig theory.
Recommended article: Combinatorial Invariance Conjecture for $\widetilde {A}_2$
Recommended article: Upper and lower bounds for Kazhdan-Lusztig polynomials
description: scientific article; zbMATH DE number 2022086