On the first coefficients in \(q\) of the Kazhdan-Lusztig polynomials
type: Item
instance of: scholarly article
title: On the first coefficients in \(q\) of the Kazhdan-Lusztig polynomials
zbMATH Open document ID: 0813.20044
author: Hiroyuki Tagawa
published in: Tokyo Journal of Mathematics
publication date: 1994-09-28T00:00:00Z
review text: A Kazhdan-Lusztig polynomial \(P_{x,w}\) with respect to two elements \(x\), \(w\) of a Coxeter group \(W\) is a certain polynomial in an indeterminate \(q\) with integer coefficients. The positivity of all the coefficients of the Kazhdan-Lusztig polynomials was conjectured by Kazhdan and Lusztig. This conjecture was verified in the cases when \(W\) is either finite or crystallographic but still remains open in general by now. In the paper under review, the author gives a combinatorial description for the coefficient \(p_ 1(x,w)\) of \(q\) in a Kazhdan-Lusztig polynomial \(P_{x,w}\) by introducing a left subword. As a result, he shows that \(p_ 1(x,w)\) is non-negative when the involved elements \(x, w \in W\) satisfy the length condition \(\ell(w) = \ell(x) + \ell(x^{-1} w)\) (resp. \(\ell(w) = \ell(x) + \ell(wx^{-1}))\).
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 645699
zbMATH Keywords: positivity of coefficients
zbMATH Keywords: Coxeter systems
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Kazhdan- Lusztig polynomials
zbMATH Keywords: left subwords
zbMATH Keywords: length condition
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270128201
OpenAlex ID: W2082194188
DOI: 10.3836/TJM/1270128201
description: scientific article; zbMATH DE number 645699