On enumerating factorizations in reflection groups

TL;DR

This paper presents a unified approach using Malle's permutation to derive formulas for counting reflection factorizations of Coxeter elements and extends to regular elements, unifying and generalizing previous results.

Contribution

It introduces a novel method based on Malle's permutation to derive and generalize formulas for reflection factorizations in reflection groups.

Findings

Derived the Chapuy-Stump formula using a new approach

Extended the formula to weighted and regular elements

Provided structural insights into factorization formulas

Abstract

We describe an approach, via Malle's permutation $\Psi$ on the set of irreducible characters $\text{Irr}(W)$, that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements.