Séminaire SPACE ToursCoxeter factorizations and the Matrix Tree theorem with generalized Jucys-Murphy weights
Theo Douvropoulos (Université Paris Diderot)
vendredi 21 février 2020 10:30 - Tours - E 1180
One of the most far reaching proofs of Cayley's result on the number of vertex-labeled trees is via Kirchhoff's Matrix Tree theorem, where the enumeration is reduced to the determinant calculation of the Laplacian L(K_n). Alon and Kozma (and independently Burman and Zvonkine) gave a remarkable generalization of this in the context of the interchange process on a graph. They exhibited a factoring formula, in terms of the eigenvalues of the graph Laplacian, for the probability that the resulting permutation is a single long cycle.
In joint work with Guillaume Chapuy, we consider a (partial) analog of the weighted Laplacian for complex reflection groups W. The weights are specified via a flag of parabolic subgroups, generalizing the definition of Jucys-Murphy elements. We prove a product formula for the enumeration of weighted reflection factorizations of Coxeter elements and in fact a Matrix Forest theorem for the characteristic polynomial of this W-Laplacian.