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

Abstract:

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation $\tau _b$ of the generating series of bipartite maps, which generalizes the partition function of $\beta$-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients $c^\lambda _{\mu ,\nu }$ of the function $\tau _b$ in the power-sum basis are non-negative integer polynomials in the deformation parameter $b$. Dołęga and Féray have proved in 2016 the “polynomiality” part in the Matching-Jack conjecture, namely that coefficients $c^\lambda _{\mu ,\nu }$ are in $\mathbb {Q}[b]$. In this paper, we prove the “integrality” part, i.e. that the coefficients $c^\lambda _{\mu ,\nu }$ are in $\mathbb {Z}[b]$.

The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums $\overline { c}^\lambda _{\mu ,l}$ from an analog result for the $b$-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra.