Representation theory of the symmetric group allows to connect Schur symmetric functions to generating series of maps on orientable surfaces. Several conjectures suggest that Jack polynomials, a one parameter deformation of Schur functions, are related to the enumeration of non-orientable maps counted with a "non-orientability" weight.
In this talk, I present an explicit formula for the power-sum expansion of Jack polynomials. This formula allows to prove a conjecture of Lassalle from 2008 on integrality and positivity of Jack characters in Stanley’s coordinates. This talk is based on a joint work with Maciej Dolega.
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