"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element

TL;DR

This paper provides a uniform combinatorial approach to count reflection factorizations of Coxeter elements in Weyl groups, improving upon previous case-by-case character-theoretic methods.

Contribution

It introduces a case-free derivation for counting reflection factorizations in Weyl groups using Deligne-Lusztig representations.

Findings

Uniform formula for reflection factorizations in Weyl groups

Simplifies previous case-by-case character evaluations

Enhances understanding of Coxeter element factorizations

Abstract

Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.