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 $(J^{(\alpha )}_\pi )$. 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 Matchings-Jack Conjecture remains open. Dołȩga and Féray showed the polynomiality of connection coefficients $c^\lambda_{\pi,\sigma}$ and gave an upper bound on the degrees. We show a dual approach to this problem and investigate Jack characters and their connection coefficients. 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.