B-Monotone Hurwitz Numbers: Virasoro Constraints, BKP Hierarchy, and O(N)-BGW Integral

Abstract

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 that generalize 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.