Jack polynomials and free cumulants

Abstract

We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group.

43 pages, LaTeX, to appear in Adv. Math

References in corpus (6)

• A recursion and a combinatorial formula for Jack polynomials
• Shifted Schur Functions
• Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
• Combinatorial interpretation and positivity of Kerov's character polynomials
• Asymptotics of characters of symmetric groups, genus expansion and free probability
• On the formula of Goulden and Rattan for Kerov polynomials

Cited by in corpus (29)

• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures
• Jack polynomials and orientability generating series of maps
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• Zonal polynomials via Stanley's coordinates and free cumulants
• Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q,t)$-Gelfand-Tsetlin Graph
• On Kerov polynomials for Jack characters
• Gaussian fluctuations of Jack-deformed random Young diagrams
• Structure coefficients for Jack characters: approximate factorization property
• Cumulants of Jack symmetric functions and $b$-conjecture
• Asymptotics of Jack characters
• Time delay statistics for finite number of channels in all symmetry classes
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Stanley character polynomials
• Asymptotic results for Representation Theory
• A Spin Analogue of Kerov Polynomials
• The top-degree part in the Matchings-Jack Conjecture
• A recurrence formula for Jack connection coefficients
• Bijection between oriented maps and weighted non-oriented maps
• Narayana polynomials and Hall-Littlewood symmetric functions
• On complete functions in Jucys-Murphy elements
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Spectral Theory of the Nazarov-Sklyanin Lax Operator
• Polynomial properties of Jack connection coefficients and generalization of a result by Dénes
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Polynomial functions on Young diagrams arising from bipartite graphs