On the Hurwitz action in affine Coxeter groups

TL;DR

This paper proves that for certain elements called parabolic quasi-Coxeter elements in affine Coxeter groups, the Hurwitz action on their factorizations into reflections is always transitive, revealing new symmetry properties.

Contribution

It establishes the transitivity of the Hurwitz action for parabolic quasi-Coxeter elements in affine Coxeter groups, a novel result in the study of Coxeter group factorizations.

Findings

Hurwitz action is transitive on factorizations of parabolic quasi-Coxeter elements

Defines parabolic quasi-Coxeter elements in affine Coxeter groups

Provides new insights into the structure of affine Coxeter groups

Abstract

We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.