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 maps (i.e., graphs drawn on surfaces), allowing to express Jack characters as weighted sums 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. This candidate weight measures somehow the non-orientability of a given map.