Zonal polynomials via Stanley's coordinates 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 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.

Mots clés: polynomes zonaux; symétriseur de Young; cumulants libres; cartes sur des surfaces non orientées

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

Valentin Féray, Piotr Sniady. Zonal polynomials via Stanley's coordinates and free cumulants. Journal of Algebra, 2011, 334, pp.338-373. ⟨hal-00480152v2⟩