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

Abstract

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