Zonal polynomials via Stanley's coordinates and free cumulants

Details Zonal polynomials via Stanley's coordinates and free cumulants Zonal polynomials via Stanley's coordinates and free cumulants

2010-05-03

arxiv.org/abs/1005.0316v2

Mathematics

Combinatorics

45 pages, second version, important changes

Scientific paper

We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanley's multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of Lassalle for Jack polynomials in the special case of zonal polynomials.

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