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arXiv · 2109.01499

$b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral

Valentin Bonzom·Guillaume Chapuy·Maciej Dołęga
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Abstract

We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We give an evolution equation for this model and derive from it Virasoro constraints, thereby proving a conjecture of F\'eray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps which generalizes monotone transposition factorizations is provided. In the case $b=1$ we obtain an explicit Schur expansion of the model and show that it obeys the BKP integrable hierarchy. This Schur expansion also proves a conjecture of Oliveira--Novaes relating zonal polynomials with irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Br\'ezin--Gross--Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.

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Valentin Bonzom, Guillaume Chapuy, Maciej Dołęga. 2021-09-03. $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral. https://doi.org/10.1093/imrn%2Frnac177

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