Reflection centralizers in Coxeter groups

TL;DR

This paper refines Brink's theorem on reflection centralizers in Coxeter groups, providing explicit generators, a method for computing Coxeter diagrams, and practical tools for understanding these structures.

Contribution

It offers an explicit set of generators for reflection centralizers and a method to compute their Coxeter diagrams, enhancing understanding and computation in Coxeter groups.

Findings

The non-reflection part of a reflection centralizer is a free group.

Provided an explicit set of generators for centralizers.

Developed a method to compute Coxeter diagrams for reflection subgroups.

Abstract

We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one's head.