A refined twist on Hurwitz numbers

TL;DR

This paper introduces a two-parameter refinement of the Jucys-Murphy theory called CJT-refinement, unifying various symmetric function actions and applying it to resolve conjectures, derive recursion formulas, and interpret Hurwitz numbers tropically.

Contribution

The paper presents the CJT-refinement, a novel framework unifying Schur, zonal, and Jack actions, with applications to Hurwitz number conjectures and tropical geometry.

Findings

Partial resolution of Coulter-Do conjecture

Derived cut-and-join recursion for b-Hurwitz numbers

Established tropicalization and polynomial structure of b-Hurwitz numbers

Abstract

We introduce a two-parameter refinement of the Jucys-Murphy theory, that we call the CJT-refinement, unifying Schur, zonal, and, conjecturally, Jack actions of the ring of symmetric functions on the Fock space. Applications of this formalism include a partial resolution of a recent conjecture of Coulter-Do, as well as cut-and-join recursion for $b$-Hurwitz numbers. The cut-and-join equations enable the derivation of the tropicalization of $b$--Hurwitz numbers. We also provide a first application of this tropical interpretation by answering an open problem of Chapuy-Do{\l}\k{e}ga on the polynomial structure of $b$-Hurwitz numbers.