On finite factors of centralizers of parabolic subgroups in Coxeter groups

TL;DR

This paper investigates the structure of centralizers of parabolic subgroups in Coxeter groups, revealing fixed point properties under certain conditions and applying these findings to the isomorphism problem.

Contribution

It provides new insights into the structure of centralizers in Coxeter groups, especially regarding finite irreducible components and their fixed points under a natural action.

Findings

Finite irreducible components are fixed by the fundamental group action under certain conditions.

The structure of centralizers is further clarified beyond known split extensions.

Application to the Coxeter group isomorphism problem is demonstrated.

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 $2 \leq n < \infty$, 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.