Parabolic subgroups of Artin groups
type: Item
instance of: scholarly article
title: Parabolic subgroups of Artin groups
zbMATH Open document ID: 0926.20022
author: Luis Paris
published in: Journal of Algebra
publication date: 1999-12-01T00:00:00Z
review text: Let \((A,\Sigma)\) be an Artin system. For \(X\subseteq\Sigma\) by \(A_X\) is denoted the subgroup of \(A\) generated by \(X\). Such a group is called a parabolic subgroup of \(A\). Van der Lek's theorem is reproved: ``A parabolic subgroup of an Artin group is an Artin group''. An algorithm is given which decides whether two parabolic subgroups of an Artin group are conjugate. Let \(A\) be a finite type Artin group and let \(A_X\) be a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer of \(A_X\) in \(A\) is the set of \(\beta\) in \(A\) such that \(\beta X\beta^{-1}=X\). It is also proved that the commensurator of \(A_X\) in \(A\) is equal to the normalizer of \(A_X\) in \(A\) and that this group is generated by \(A_X\) and the quasi-centralizer of \(A_X\) in \(A\). Moreover, if \(A_X\) is not of type \(D_l\) (\(l\geq 4\) and \(l\) even) then this group is generated by \(A_X\) and the centralizer of \(A_X\) in \(A\).
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1088589
zbMATH Keywords: Artin systems
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: Artin groups
zbMATH Keywords: quasi-centralizers
zbMATH Keywords: commensurators
reviewed by: Milica Stojanović
MaRDI profile type: Publication
full work available at URL: jabr.1997.7098
OpenAlex ID: W2043357595
cites work: Artin groups and infinite Coxeter groups
cites work: Q5580319
cites work: Artin-Gruppen und Coxeter-Gruppen
cites work: Q4693148
cites work: The K(π, 1)-Problem for Hyperplane Complements Associated to Infinite Reflection Groups
cites work: The Complete Enumeration of Finite Groups of the Form Ri2=(RiRj)kij=1
cites work: Nonpositive curvature and reflection groups
cites work: Les immeubles des groupes de tresses généralises
cites work: On the root system of a coxeter group
cites work: Centralisers in the braid group and singular braid monoid
cites work: THE BRAID GROUP AND OTHER GROUPS
cites work: Q4865837
cites work: Normalizers of Parabolic Subgroups of Reflection Groups
cites work: Q3998721
cites work: The conjugacy problem for Coxeter groups.
cites work: Q3882652
cites work: Commensurators of parabolic subgroups of Coxeter groups
cites work: Conjugacy classes of involutions in Coxeter groups
cites work: Braid subgroup normalisers, commensurators and induced representations
cites work: The homotopy type of Artin groups
cites work: Q5605923
DOI: 10.1006/JABR.1997.7098
description: scientific article; zbMATH DE number 1088589