We demonstrate cut-and-join recursion for Hurwitz numbers in Fock space, suggesting a unifying theory.
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-Dołęga on the polynomial structure of b-Hurwitz numbers.
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Fesler et al. (2025) studied this question.
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