On Kerov polynomials for Jack characters

Abstract

We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.

arXiv commentThis version is an extended abstract (12 pages, 1 figure) of arXiv:1402.4615 with significantly fewer results and a different title

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• Polynomial properties of Jack connection coefficients and generalization of a result by Dénes
• A multivariate hook formula for labelled trees