Kazhdan-Lusztig polynomials of parabolic type
type: Item
instance of: scholarly article
title: Kazhdan-Lusztig polynomials of parabolic type
zbMATH Open document ID: 0899.20020
author: Hiroyuki Tagawa
published in: Journal of Algebra
publication date: 1998-11-16T00:00:00Z
review text: Let \((W,S)\) be a Coxeter system. For \(J\subseteq S\), let \(W_J\) be the subgroup of \(W\) generated by \(J\) and let \(W^J\) be the set of all the elements \(w\in W\) such that \(w\) has the minimal length in the coset \(wW_J\). An \(R\)-polynomial \({\mathcal R}^J_{x,w}(q)_u\) of parabolic type (resp. a Kazhdan-Lusztig polynomial \(P^J_{x,w}(q)_u\) of parabolic type) is a certain polynomial in an indeterminate \(q\) indexed by two elements \(x,w\in W^J\), where \(u=-1\) or \(u=q\). The positivity of the coefficients of the polynomials \(P^J_{x,w}(q)_u\) is a long-standing conjecture. It has been shown only when \(W\) is a crystallographic Coxeter group. The author gives a formula for the polynomials \(P^J_{x,w}(q)_u\) in terms of \(R\)-polynomials. This formula has a pure combinatorial nature since the \({\mathcal R}_{x,w}^J(q)_u\)'s are defined in terms of multichains in \(W^J\) under the Bruhat order. The author applies the formula to show the non-negativity of the coefficient of \(q\) in \(P^J_{x,w}(q)_u\) under a certain condition on the length of elements. Note that the formula for the polynomials \(P^J_{x,w}(q)_u\) was also obtained independently by \textit{V. Deodhar} [see J. Algebra 190, No. 1, 214-225 (1997; Zbl 0889.20021)].
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1139778
zbMATH Keywords: \(R\)-polynomials
zbMATH Keywords: Coxeter systems
zbMATH Keywords: Kazhdan-Lusztig polynomials
zbMATH Keywords: positivity of coefficients
zbMATH Keywords: crystallographic Coxeter groups
zbMATH Keywords: multichains
zbMATH Keywords: Bruhat order
MaRDI profile type: Publication
full work available at URL: jabr.1997.7220
OpenAlex ID: W2037925367
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cites work: A combinatorial formula for Kazhdan-Lusztig polynomials
cites work: On some geometric aspects of Bruhat orderings. II: The parabolic analogue of Kazhdan-Lusztig polynomials
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cites work: Q4338604
cites work: Q3998721
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q3748279
cites work: On the first coefficients in \(q\) of the Kazhdan-Lusztig polynomials
cites work: On the non-negativity of the first coefficient of Kazhdan-Lusztig polynomials
DOI: 10.1006/JABR.1997.7220
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description: scientific article; zbMATH DE number 1139778