Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra. (English) Zbl 1512.05392
Summary: Using Jack polynomials, I. P. Goulden and D. M. Jackson [J. Algebra 166, No. 2, 364–378 (1994; Zbl 0830.20021)] 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 \(b\). M. Dołęga and V. Féray [Duke Math. J. 165, No. 7, 1193–1282 (2016; Zbl 1338.60017)] have proved 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 [G. Chapuy and M. Dołęga, Adv. Math. 409 A, Article ID 108645, 72 p. (2022; Zbl 1498.05278)]. A key step in the proof involves a new connection with the graded Farahat-Higman algebra.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 15B52 | Random matrices (algebraic aspects) |
| 20C30 | Representations of finite symmetric groups |
Keywords:
symmetric functions; connexion coefficients; class algebras; symmetric groups; ordered factorisations of permutationsReferences:
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