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

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 Féray 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 Brézin--Gross--Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.

v2: 44 pages. v3: added an assumption in Lemma 2.6

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• On the Complex Asymptotics of the HCIZ and BGW Integrals
• WKB-expansion of the HarishChandra-Itzykson-Zuber integral for arbitrary beta

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• Hurwitz numbers for reflection groups III: Uniform formulas