A Note on Element Centralizers in Finite Coxeter Groups

TL;DR

This paper investigates the structure of element centralizers in finite Coxeter groups, revealing they are often semidirect products of smaller parabolic subgroup centralizers and normalizer complements, with some exceptions.

Contribution

It extends understanding of centralizer structures in finite Coxeter groups, showing they are typically semidirect products similar to normalizer decompositions, except in certain type D cases.

Findings

Centralizers are often semidirect products of smaller parabolic centralizers and normalizer complements.

Most cases follow a similar splitting pattern, with notable exceptions in type D Coxeter groups.

Provides a detailed structural description of element centralizers in finite Coxeter groups.

Abstract

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.