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

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