A note on non-reduced reflection factorizations of Coxeter elements

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.

The Hurwitz action in complex reflection groups

“…The analogous result was proved for (not necessarily finite) Coxeter groups by Igusa and Schiffler [15], with a short and self-contained proof in [1]. This has been extended to arbitrary-length reflection factorizations of Coxeter elements in Coxeter groups (first finite [21], then in general [34]) and to the infinite family of complex reflection groups [23]. Finally, in [33], Wegener extended one direction of the main result of [2] to the case of affine Coxeter groups.…”

Hurwitz Orbits on Reflection Factorizations of Parabolic Quasi-Coxeter Elements

“…We will make use of the following structural lemma for the Hurwitz action on reflection factorizations throughout the paper. The proof given in [LR16] relies on the classification of finite Coxeter groups, but recently Wegener and Yahiatene gave a uniform proof [WY23].…”

On the Hurwitz action in affine Coxeter groups

“…For Coxeter elements, the transitivity was first shown by Igusa and Schiffler [IS10]. The Hurwitz action in Coxeter groups has also been studied for nonreduced reflection decompositions [LR16,WY] and outside of the context of parabolic Coxeter elements [Voi85,Mic06].…”