Automorphisms of graph-universal Coxeter groups
type: Item
instance of: scholarly article
title: Automorphisms of graph-universal Coxeter groups
zbMATH Open document ID: 0897.20033
author: Bernhard Mühlherr
published in: Journal of Algebra
publication date: 1998-05-25T00:00:00Z
review text: This paper deals with the group \(\Aut(W)\) of automorphisms of a graph-universal type Coxeter group. Recall that a Coxeter diagram \(M=(m_{i,j})_{i,j\in I}\) over a set \(I\) is called graph-universal if \(m_{i,j}\in\{2,\infty\}\) for all \(i,j\in I\). Consider a Coxeter group \(W\) associated with such a diagram. Then \(W\) acts on the conjugacy classes of involutions of \(W\). Let \(\text{Spe}(W)\) denote the kernel of this action, and let \(\text{Inn}(W)\) denote the group of inner automorphisms of \(W\). It was shown by \textit{J. Tits} [J. Algebra 113, No. 2, 346-357 (1988; Zbl 0657.20027)]\ that \(\Aut(W)\) is a semidirect extension of \(\text{Spe}(W)\) with some finite group. In this paper, the author gives an explicit presentation of \(\text{Spe}(W)\), and deduces a necessary and sufficient condition for \(\text{Spe}(W)\) to be equal to \(\text{Inn}(W)\).
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1139793
zbMATH Keywords: Coxeter diagrams
zbMATH Keywords: Coxeter groups
zbMATH Keywords: conjugacy classes of involutions
zbMATH Keywords: groups of inner automorphisms
zbMATH Keywords: presentations
reviewed by: Luis Paris
MaRDI profile type: Publication
full work available at URL: jabr.1997.7230
OpenAlex ID: W2065333242
cites work: Q5580319
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cites work: Conjugacy classes of involutions in Coxeter groups
cites work: Q4692803
cites work: Q5605923
cites work: Sur le group des automorphismes de certains groupes de Coxeter. (On the automorphism group of certain Coxeter groups)
DOI: 10.1006/JABR.1997.7230
description: scientific article; zbMATH DE number 1139793