Hurwitz numbers for reflection groups $G(m,1,n)$

Abstract

We build a parallel theory of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We study analogs of the cut-and-join operators. An algebraic description as well as a description of Hurwitz numbers in terms of ramified coverings is provided. An explicit formula for them in terms of Schur polynomials is given. In addition, the generating function of $G(m,1,n)$-Hurwitz numbers is shown to give rise to an $m$ independent-variables $\tau$-function of the KP hierarchy. Finally, we provide an ELSV-type formula for these new Hurwitz numbers. These results extend the results of Fesler (2023).