In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants

TL;DR

This paper proves that for parabolic quasi-Coxeter elements in finite real reflection groups, the Hurwitz orbits of their reflection factorizations are uniquely identified by subgroup generation and conjugacy class multisets.

Contribution

It establishes a complete characterization of Hurwitz orbits for these elements and classifies certain minimal reflection generating sets in finite Coxeter groups.

Findings

Hurwitz orbits are distinguished by subgroup and conjugacy class invariants

Classification of finite Coxeter groups with minimal reflection generating sets

Proof of the equivalence condition for reflection factorizations

Abstract

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.