On the Hurwitz action in finite Coxeter groups

TL;DR

This paper characterizes when the Hurwitz action is transitive on reduced decompositions of elements in finite Coxeter groups, identifying parabolic quasi-Coxeter elements as the key condition.

Contribution

It provides a necessary and sufficient condition for the transitivity of the Hurwitz action in finite Coxeter groups, linking it to parabolic quasi-Coxeter elements.

Findings

Hurwitz action is transitive iff the element is a parabolic quasi-Coxeter element.

Characterization of elements with transitive Hurwitz action in finite Coxeter groups.

Connection between reduced decompositions and parabolic subgroups.

Abstract

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.