Hurwitz numbers for reflection groups $B$ and $D$

Abstract

We are building a theory of simple Hurwitz numbers for the reflection groups B and D parallel to the classical theory for the symmetric group. We also study analogs of the cut-and-join operators. An algebraic description of Hurwitz numbers and an explicit formula for them in terms of Schur polynomials are provided. We also relate Hurwitz numbers for B and D to ribbon decomposition of surfaces with boundary -- a similar result for the symmetric group was proved earlier by Yu.Burman and the author. Finally, the generating function of B-Hurwitz numbers is shown to give rise to two independent tau-function of the KP hierarchy.