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arXiv · 2508.06188

A refined twist on Hurwitz numbers

Raphaël Fesler·Marvin Anas Hahn·Maksim Karev·Hannah Markwig
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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.

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Raphaël Fesler, Marvin Anas Hahn, Maksim Karev, Hannah Markwig. 2025-08-08. A refined twist on Hurwitz numbers. https://arxiv.org/abs/2508.06188

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