Centralizers of reflections and reflection-independence of Coxeter groups

TL;DR

This paper introduces a new criterion for reflection independence in Coxeter groups, explores its applicability to various classes, and characterizes certain subgroups related to centralizing reflections, enhancing understanding of Coxeter group structure.

Contribution

It provides a novel sufficient condition for reflection independence and applies it to specific Coxeter groups, including infinite rank cases, while also characterizing related centralizer subgroups.

Findings

New criterion for reflection independence established

Finite irreducible components of specific subgroups determined

Criterion shown to be effective for infinite rank Coxeter groups

Abstract

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this condition for Coxeter groups in certain classes, possibly of infinite ranks. We also determine the finite irreducible components of another Coxeter group, that is a subgroup of W generated by the reflections centralizing a given generator of W. Determining such a subgroup makes our criterion efficient.