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 $W$, in terms of the spectrum of an associated operator, the $W$-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 $n$ 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 $W$. Finally we study how this $W$-Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics. We explain how it defines analogues of trees for $W$ and how it relates them to Coxeter factorizations, we give new numerological identities between the Coxeter number of $W$ and those of its parabolic subgroups, and finally, when $W$ is a Weyl group, we produce a new, explicit formula for the volume of the corresponding root zonotope.

arXiv commentv2: 59 pages, Lemma 7.2 corrected

References in corpus (10)

• Proof of Aldous' spectral gap conjecture
• Zariski theorems and diagrams for braid groups
• Affine and toric hyperplane arrangements
• Counting factorizations of Coxeter elements into products of reflections
• On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
• Reflection factorizations of Singer cycles
• The probability of long cycles in interchange processes
• Matrix-Forest Theorems
• Counting chains in the noncrossing partition lattice via the W-Laplacian
• Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth)

Cited by in corpus (5)

• Counting chains in the noncrossing partition lattice via the W-Laplacian
• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• The diagonal coinvariant ring of a complex reflection group
• Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
• The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces