Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra

Abstract

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformationτb\tau _bof the generating series of bipartite maps, which generalizes the partition function ofβ\beta-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficientscμ,νλc^\lambda _{\mu ,\nu }of the functionτb\tau _bin the power-sum basis are non-negative integer polynomials in the deformation parameterbb. Dołęga and Féray have proved in 2016 the “polynomiality” part in the Matching-Jack conjecture, namely that coefficientscμ,νλc^\lambda _{\mu ,\nu }are inQ[b]\mathbb {Q}[b]. In this paper, we prove the “integrality” part, i.e. that the coefficientscμ,νλc^\lambda _{\mu ,\nu }are inZ[b]\mathbb {Z}[b].The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sumsc¯μ,lλ\overline { c}^\lambda _{\mu ,l}from an analog result for thebb-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra.