Bases for the Bruhat-Chevalley order on all finite Coxeter groups
type: Item
instance of: scholarly article
title: Bases for the Bruhat-Chevalley order on all finite Coxeter groups
zbMATH Open document ID: 0977.20033
author: Meinolf Geck
author: Sungsoon Kim
published in: Journal of Algebra
publication date: 2002-01-21T00:00:00Z
review text: According to the definition by \textit{A. Lascoux} and \textit{M. P. Schützenberger}, a base for the Bruhat-Chevalley order \(\leq\) on a Coxeter group \(W\) is a minimal subset \(B\) of \(W\) such that if \(({\mathcal P}(B),\subseteq)\) is the partially ordered set of all subsets of \(B\) then the map \(W\to{\mathcal P}(B)\), \(w\mapsto\{b\in B\mid b\leq w\}\), is an isomorphism of partially ordered sets onto its image [see Electron. J. Combin. 3, No.~2, R27 (1996) = J. Comb. 3, No.~2, 633-667 (1996; Zbl 0885.05111)]. The existence of a base allows us to transform the Bruhat-Chevalley order relation to a purely Boolean condition, the latter provides an efficient and practical criterion for determining the relation \(v\leq w\) in \(W\). An element \(w\in W\) is called a bi-Grassmannian if each of the sets \({\mathcal L}(w)=\{s\in S\mid\ell(sw)<\ell(w)\}\), \({\mathcal R}(w)=\{s\in S\mid\ell(ws)<\ell(w)\}\) consists of precisely one element. The basic result of [loc. cit.] states that the base \(B\) of \((W,\leq)\) is contained in the set of all bi-Grassmannians of \(W\). In the present paper, the authors determine the base for any finite Coxeter group \(W\). In doing this, the authors adopt another (equivalent) definition of the base \(B\) of \(W\) in [loc. cit.]: \(B\) is the set of all elements \(w\in W\) which cannot be obtained as the supremum of a subset of \(W\) not containing \(w\). The above result of [loc. cit.] is crucial in the present work. In the classical cases (i.e., types \(A\), \(B\) and \(D\)), the authors use a coding of the elements of \(W\), which is given in terms of products of minimal right coset representatives along a naturally chosen chain of parabolic subgroups of \(W\). In the exceptional cases, they implement an algorithm in the computer algebra systems GAP and CHEVIE.
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 1097709
zbMATH Keywords: Bruhat-Chevalley orders
zbMATH Keywords: partially ordered sets
zbMATH Keywords: bi-Grassmannians
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: right coset representatives
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: algorithms
zbMATH Keywords: computer algebra systems
describes a project that uses: CHEVIE
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: jabr.1997.7096
OpenAlex ID: W1986201476
cites work: Q5580319
cites work: Some characterizations of Bruhat ordering on a Coxeter group and determination of the relation Möbius function
cites work: CHEVIE -- A system for computing and processing generic character tables
cites work: Q3998721
cites work: Characters of Reductive Groups over a Finite Field. (AM-107)
cites work: Classical Bruhat orders and lexicographic shellability
DOI: 10.1006/JABR.1997.7096
description: scientific article; zbMATH DE number 1097709