Asymptotics of Jack characters

Abstract

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack characters from the viewpoint of the asymptotic representation theory. In particular, we give explicit formulas for their asymptotically top-degree part, in terms of bicolored oriented maps with an arbitrary face structure. We also study their multiplicative structure and their structure constants and we prove that they fulfill approximate factorization property, a convenient tool for proving Gaussianity of fluctuations of random Young diagrams.

52 pages. Version 3: change of title. This version was created by merging two papers (and removing a lot of the discussion): arXiv:1603.04268 and version 2 of the current paper

References in corpus (6)

• Gaussian fluctuations of Young diagrams and structure constants of Jack characters
• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• Jack polynomials and free cumulants
• Jack polynomials and orientability generating series of maps
• Gaussian fluctuations of Jack-deformed random Young diagrams
• The top-degree part in the Matchings-Jack Conjecture

Cited by in corpus (6)

• Time delay statistics for finite number of channels in all symmetry classes
• Random strict partitions and random shifted tableaux
• Linear versus spin: representation theory of the symmetric groups
• A Spin Analogue of Kerov Polynomials
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures