A refined count of Coxeter element factorizations
Elise delMas, Thomas Hameister, and Victor Reiner

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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · semigroups and automata theory
