An abstract model for Bruhat intervals
type: Item
instance of: scholarly article
title: An abstract model for Bruhat intervals
zbMATH Open document ID: 0953.05083
author: Fokko du Cloux
published in: European Journal of Combinatorics
publication date: 2001-01-23T00:00:00Z
review text: Let \((W,S)\) be a Coxeter system. It has been shown by \textit{V. Deodhar} [Invent. Math. 39, 187--198 (1977; Zbl 0333.20041)] that the Bruhat ordering in \(W\) can be characterized in terms of some properties of its intervals. In the paper under review these properties are axiomatized and the abstract model for Bruhat intervals is studied. It is shown that the corresponding posets are always Eulerian and that chain complexes of their proper parts \(]\hat{0},\hat{1}[\) are triangulations of spheres. In the last section an attempt is made to define analogues of the Kazhdan-Lusztig polynomials in the abstract context. Although examples show that it is not possible in full generality, the author was able to reprove for a special class of adihedral posets a result of \textit{F.~Brenti} [Invent. Math. 118, No. 2, 371--394 (1994; Zbl 0836.20054)] stating that for an adihedral Bruhat interval the Kazhdan-Lusztig polynomial is equal to the Stanley \(g\)-polynomial of the dual interval.
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 52B22
zbMATH DE Number: 1423288
zbMATH Keywords: Coxeter group
zbMATH Keywords: Bruhat ordering
zbMATH Keywords: Kazhdan-Lusztig polynomial
zbMATH Keywords: shellability
zbMATH Keywords: Eulerian poset
reviewed by: Valeriy B. Mnukhin
MaRDI profile type: Publication
OpenAlex ID: W1971763813
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cites work: Möbius inversion for the Bruhat ordering on a Weyl group
DOI: 10.1006/EUJC.1999.0343
description: scientific article; zbMATH DE number 1423288