Zonal polynomials via Stanley-Féray formula and free cumulants

Valentin Féray (1) , Piotr Sniady (2, 3)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 INSTITUTE OF MATHEMATICS
3 INSTITUTE OF MATHEMATICS

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 zonal characters are explicitly given by an analogue of Stanley-Féray formula for character values of symmetric groups. We also study an analogue of Kerov polynomials, namely we express 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.

Mots clés: polynomes zonaux; symétriseur de Young; cumulants libres

Domaines: Combinatoire [math.CO]; Théorie des représentations [math.RT]

Valentin Féray, Piotr Sniady. Zonal polynomials via Stanley-Féray formula and free cumulants. 2010. ⟨hal-00480152v1⟩