A refined count of Coxeter element factorizations

TL;DR

This paper refines the counting of reflection factorizations of Coxeter elements in well-generated complex reflection groups, providing new formulas and case-free proofs for related combinatorial and algebraic properties.

Contribution

It introduces a refined enumeration method for reflection factorizations, incorporating hyperplane orbit data, and offers a case-free proof of a new Coxeter number expression.

Findings

Refined counting formula for reflection factorizations by hyperplane orbit

New expression for the Coxeter number using hyperplane data

Case-free proof of Michel's Coxeter number formula

Abstract

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.