Jack polynomials and orientability generating series of maps

TL;DR

This paper explores the connection between Jack characters and non-oriented maps, proposing a candidate weight function that partially expresses Jack characters as sums over maps, linking algebraic and topological properties.

Contribution

It introduces a conjecture relating Jack characters to weighted sums over non-oriented maps and proposes a candidate weight function that supports this conjecture in specific cases.

Findings

Candidate weight function supports the conjecture for rectangular Young diagrams.

The proposed weight measures non-orientability of maps.

Partial positive results suggest a link between Jack characters and map topology.

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.