Counting factorizations of Coxeter elements into products of reflections

TL;DR

This paper derives a uniform product formula for counting factorizations of Coxeter elements into reflections in well-generated complex reflection groups, generalizing known results for real cases and symmetric groups.

Contribution

It introduces a simple, uniform product formula for the exponential generating function of such factorizations across various complex reflection groups.

Findings

Derived a uniform product formula for the generating function

Recovered known formulas for minimal length factorizations

Specialized the formula to the symmetric group case

Abstract

In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.