On Kerov polynomials for Jack characters

Details On Kerov polynomials for Jack characters On Kerov polynomials for Jack characters

2012-01-09

arxiv.org/abs/1201.1806v1

Mathematics

Combinatorics

37 pages, 4 figures

Scientific paper

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, a small part of Matching Jack conjecture from I. P. Goulden and D. M. Jackson is proved. Thirdly, we describe asymptotically the shape of random Young diagrams under Jack deformation of Plancherel measure.

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