Involutions and reflection subgroups of finite Coxeter groups.
type: Item
instance of: scholarly article
title: Involutions and reflection subgroups of finite Coxeter groups.
zbMATH Open document ID: 1191.20037
author: Paul Moszkowski
published in: European Journal of Combinatorics
publication date: 2009-12-03T00:00:00Z
review text: This paper, published after the author's death in 2004, gives a ``purely algebraic'' derivation of the classification of the finite Coxeter groups -- namely those finite groups generated by involutions \(s_i\) subject to defining relations \({s_i}^2=(s_is_j)^{m_{ij}}=1\) where the \(m_{ij}\) are specified integers. The approach, which involves a fairly detailed analysis of multiplication of words in the generators, is not surprising; other authors have indicated how this approach (or aspects of it) would work. A more standard and much easier proof of the classification uses geometric representations. Here the classification depends on Proposition 2, which asserts that the order of each product \(s_is_j\) of a pair of generators is equal to the exponent \(m_{ij}\) given in the group presentation, or is infinite when \(m_{ij}\) is unspecified. This is proved in an addendum [see Eur. J. Comb. 30, No. 8, 1913-1918 (2009; Zbl 1191.20038)].
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 5641715
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: finite reflection groups
zbMATH Keywords: products of generators
reviewed by: Marston D. E. Conder
MaRDI profile type: Publication
full work available at URL: j.ejc.2009.02.002
OpenAlex ID: W1988945890
cites work: Q5580319
cites work: Q4693148
cites work: Discrete groups generated by reflections
cites work: The Complete Enumeration of Finite Groups of the Form Ri2=(RiRj)kij=1
cites work: Reflection subgroups of Coxeter systems
cites work: Q5336354
cites work: Q5605923
DOI: 10.1016/J.EJC.2009.02.002
description: scientific article; zbMATH DE number 5641715