Zonal polynomials via Stanley's coordinates and free cumulants

Abstract

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.

45 pages, second version, important changes

References in corpus (9)

• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• Jack polynomials and free cumulants
• Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
• Irreducible Symmetric Group Characters of Rectangular Shape
• Proof of Stanley's conjecture about irreducible character values of the symmetric group
• Combinatorial interpretation and positivity of Kerov's character polynomials
• A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group
• Asymptotics of characters of symmetric groups related to Stanley character formula
• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

Cited by in corpus (11)

• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Jack polynomials and orientability generating series of maps
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices
• Asymptotics of Jack characters
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Stanley character polynomials
• Asymptotic results for Representation Theory
• Linear versus spin: representation theory of the symmetric groups
• A Spin Analogue of Kerov Polynomials
• Bijection between oriented maps and weighted non-oriented maps