$b$-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.

arXiv commentv2: 44 pages. v3: added an assumption in Lemma 2.6

References in corpus (23)

• Hurwitz numbers and intersections on moduli spaces of curves
• An algebro-geometric proof of Witten's conjecture
• Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model
• Free fermions and tau-functions
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Cut-and-join description of generalized Brezin-Gross-Witten model
• Hodge integrals and Hurwitz numbers via virtual localization
• Fermionic representation for basic hypergeometric functions related to Schur polynomials
• Jucys-Murphy elements and Weingarten matrices
• Generating functions for weighted Hurwitz numbers
• Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures
• Monotone Hurwitz numbers in genus zero
• An extension of the HarishChandra-Itzykson-Zuber integral
• HCIZ integral and 2D Toda lattice hierarchy
• On the Complex Asymptotics of the HCIZ and BGW Integrals
• Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
• Some properties of angular integrals
• Cut-and-join equation for monotone Hurwitz numbers revisited
• Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model
• WKB-expansion of the HarishChandra-Itzykson-Zuber integral for arbitrary beta
• On the immanants of blocks from random matrices in some unitary ensembles

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• Refined topological recursion revisited -- properties and conjectures
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• Enumeration of non-oriented maps via integrability
• Jacobi Beta Ensemble and $b$-Hurwitz Numbers
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion
• Hurwitz numbers for reflection groups III: Uniform formulas
• $\beta$-ensembles and higher genera Catalan numbers