Cohomology of Coxeter groups and Artin groups
type: Item
instance of: scholarly article
title: Cohomology of Coxeter groups and Artin groups
zbMATH Open document ID: 0972.20030
published in: Mathematical Research Letters
publication date: 2001-08-22T00:00:00Z
review text: For an irreducible Coxeter system \((W,S)\), with the group \(W\) finite, the authors construct an explicit free resolution \((C_*,\delta_*)\) of the trivial \(\mathbb{Z}[W]\)-module \(\mathbb{Z}\). In dimension \(k\), \(C_k\) is the free \(\mathbb{Z}[W]\)-module on the flags of subsets of \(S\) of cardinality \(k\). If \(n\) is the rank of \(W\), then there are \({n+k-1}\choose k\) such flags, so this resolution is much smaller than the standard resolution. Examples of calculations of cohomology using the resolution are included. The authors raise the question of when their resolution is the smallest possible, and this was answered by \textit{I. J. Leary} in his review of the article [in MR 2001f:20118].
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 1506008
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Artin groups
zbMATH Keywords: free resolutions
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: flags
zbMATH Keywords: arrangements
zbMATH Keywords: chambers
zbMATH Keywords: posets
zbMATH Keywords: cohomology
zbMATH Keywords: Schwarz genus
zbMATH Keywords: Specht modules
author: Corrado De Concini
author: Mario Salvetti
MaRDI profile type: Publication
DOI: 10.4310/MRL.2000.V7.N2.A7
description: scientific article; zbMATH DE number 1506008