Séminaire Lotharingien de Combinatoire, 84B.82 (2020), 12 pp.
Guillaume Chapuy and Theo Douvropoulos
Coxeter factorizations and the Matrix Tree theorem with generalized Jucys-Murphy weights
We prove a universal (case-free) formula for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group, in terms of the spectrum of an associated Laplacian matrix that we introduce. This covers in particular all finite Coxeter groups. For symmetric groups and for minimal length, our statement is an instance of the Matrix Tree theorem.
The formula is relative to the choice of a weighting system, that corresponds to the choice of $n$ free scalar parameters and of a parabolic tower of subgroups. This leads us to 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. The main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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