Jack polynomials and orientability generating series of maps

Abstract

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.

v2: change of title, substantial changes of the content v3: substantial changes in the presentation

References in corpus (5)

• Fractional Quantum Hall States and Jack Polynomials
• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Jack polynomials and free cumulants
• A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group
• On Kerov polynomials for Jack characters

Cited by in corpus (13)

• A bijection for rooted maps on general surfaces
• Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
• Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q,t)$-Gelfand-Tsetlin Graph
• Cumulants of Jack symmetric functions and $b$-conjecture
• Asymptotics of Jack characters
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Stanley character polynomials
• A recurrence formula for Jack connection coefficients
• Linear versus spin: representation theory of the symmetric groups
• A Spin Analogue of Kerov Polynomials
• Polynomial properties of Jack connection coefficients and generalization of a result by Dénes
• Jack characters and enumeration of maps
• Algebras with two multiplications and their cumulants