On finite factors of centralizers of parabolic subgroups in Coxeter groups

Abstract

It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathcal{Y}$. In this paper, we study the structure of $Z_W(W_I)$ further and show that, if $I$ has no irreducible components of type $A_n$ with $2\le n<\infty$, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $\mathcal{Y}$. This property has an application to the isomorphism problem in Coxeter groups.

44 pages, 4 figures, 19 tables, separated from Section 7 of arXiv:math/0501061v4 [math.GR] with improvements; (v2) 43 pages, 5 figures, 16 tables, the size of counterexample is reduced, some parts are slightly revised

References in corpus (2)

• On centralizers of parabolic subgroups in Coxeter groups
• Centralizers of reflections and reflection-independence of Coxeter groups