A note on non-reduced reflection factorizations of Coxeter elements

TL;DR

This paper generalizes a known result about reflection factorizations of Coxeter elements from finite to all Coxeter groups, establishing a criterion for when two factorizations are in the same Hurwitz orbit based on conjugacy classes.

Contribution

It extends the classification of reflection factorizations of Coxeter elements to infinite Coxeter groups, broadening the scope of previous finite group results.

Findings

Two reflection factorizations are in the same Hurwitz orbit iff they share the same multiset of conjugacy classes.

The result applies to all Coxeter groups, not just finite ones.

Provides a unified framework for understanding factorizations across different Coxeter groups.

Abstract

We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.