Coxeter factorizations with generalized Jucys–Murphy weights and Matrix‐Tree theorems for reflection groups

Abstract

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group W $W$ , in terms of the spectrum of an associated operator, the W $W$ -Laplacian. This covers in particular all finite Coxeter groups. The results of this paper include generalizations of the Matrix-Tree and Matrix Forest theorems to reflection groups, and cover reduced (shortest length) as well as arbitrary length factorizations. Our formulas are relative to a choice of weighting system that consists of n $n$ free scalar parameters and is defined in terms of a tower of parabolic subgroups. To study such systems we introduce (a class of) variants of the Jucys–Murphy elements for every group, from which we define a new notion of “tower equivalence” of virtual characters. A main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation of W $W$ . Finally we study how this W $W$ -Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics; for instance, we explain how it defines analogs of trees for W $W$ and how it relates them to Coxeter factorizations. Moreover, we build a parabolic recursion for the W $W$ -Laplacian that proves, without relying on the classification, new numerological identities between the Coxeter number of W $W$ and those of its parabolic subgroups, and, when W $W$ is a Weyl group, a new, explicit formula for the volume of the corresponding root zonotope.