Circuits and Hurwitz action in finite root systems

TL;DR

This paper characterizes when two factorizations of a Coxeter element are related by the Hurwitz action in finite real reflection groups, linking this to the multiset of conjugacy classes used.

Contribution

It establishes a criterion for Hurwitz equivalence of factorizations based on conjugacy class multisets, using a novel lemma about root system dependencies.

Findings

Hurwitz action orbits correspond to conjugacy class multisets

Minimal linear dependencies have disconnected acuteness graphs

Provides a classification tool for factorizations in finite root systems

Abstract

In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.