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

TL;DR

This paper develops a new theory of Hurwitz numbers for reflection groups B and D, providing algebraic formulas, surface interpretations, and connections to integrable systems, extending classical results for the symmetric group.

Contribution

It introduces a novel framework for Hurwitz numbers associated with groups B and D, including explicit formulas and surface decompositions, expanding the scope of classical Hurwitz theory.

Findings

Algebraic description of Hurwitz numbers for B and D

Explicit formulas in terms of Schur polynomials

Connection to tau-functions of the KP hierarchy

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.