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Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions (extended abstract)

Abstract : We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the "$b$-conjecture" of Goulden and Jackson from 1996, expansions of the $\beta$-ensemble matrix model, deformations of the HCIZ integral, and $b$-Hurwitz numbers that we introduce here and that are $b$-deformations of classical (single or double) Hurwitz numbers obtained for $b=0$. A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable $b$-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.
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https://hal.archives-ouvertes.fr/hal-03379706
Contributor : Guillaume Chapuy Connect in order to contact the contributor
Submitted on : Friday, October 15, 2021 - 10:15:46 AM
Last modification on : Sunday, October 17, 2021 - 3:13:27 AM

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  • HAL Id : hal-03379706, version 1
  • ARXIV : 2004.07824

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Guillaume Chapuy, Maciej Dołęga. Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions (extended abstract). FPSAC 2021 -- Sém. Lothar. Combin. 85B (2021), Art. 32, 13 pp., Jan 2022, Ramat-Gan, Israel. ⟨hal-03379706⟩

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