The top-degree part in the Matchings-Jack Conjecture

Abstract

In 1996 Goulden and Jackson introduced a family of coefficients $(c_{\pi ,\sigma }^{\lambda })$ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $(J_{\pi }^{(\alpha )})$. The coefficients $c_{\pi ,\sigma }^{\lambda }$ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $c_{\pi ,\sigma }^{\lambda }$ are polynomials in the variable $\beta :=\alpha -1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This \emph{Matchings-Jack Conjecture} remains open. Doł\oldk{e}ga and Féray showed the polynomiality of connection coefficients $c_{\pi ,\sigma }^{\lambda }$ and gave the upper bound on the degrees. We give a necessary and sufficient condition for the polynomial $c_{\pi ,\sigma }^{\lambda }$ to achieve this bound. We show that the leading coefficient of $c_{\pi ,\sigma }^{\lambda }$ is a positive integer and we present it in the context of Matchings-Jack Conjecture of Goulden and Jackson.

34 pages

References in corpus (7)

• Jack polynomials and free cumulants
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• The uses of random partitions
• Cumulants of Jack symmetric functions and $b$-conjecture
• Structure coefficients for Jack characters: approximate factorization property
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• On the matchings-Jack and hypermap-Jack conjectures for labelled matchings and star hypermaps

Cited by in corpus (1)

• Algebras with two multiplications and their cumulants