Jack Characters as Generating Series of Bipartite Maps and Proof of Lassalle's Conjecture

Explore a 50-minute lecture on Jack characters and their connection to bipartite maps, presented by Houcine Ben Dali from the Université de Lorraine at IPAM's Integrability and Algebraic Combinatorics Workshop. Delve into the relationship between representation theory of the symmetric group and generating series of maps on orientable surfaces. Discover how Jack polynomials, a one-parameter deformation of Schur functions, are linked to the enumeration of non-orientable maps with a "non-orientability" weight. Examine an explicit formula for the power-sum expansion of Jack polynomials, which leads to the proof of Lassalle's 2008 conjecture on the integrality and positivity of Jack characters in Stanley's coordinates. Gain insights into this collaborative work with Maciej Dolega, advancing our understanding of algebraic combinatorics and representation theory.