Bijection between oriented maps and weighted non-oriented maps

TL;DR

This paper establishes a bijection between oriented and weighted non-oriented maps on surfaces, preserving the underlying graph, and demonstrates its implications for Jack character formulas.

Contribution

It introduces a novel bijection linking oriented and non-oriented maps, revealing their equivalence in computing Jack character coefficients.

Findings

Bijection preserves the underlying bicolored graph.

Shows equivalence between two formulas for Jack characters.

Connects map topology with symmetric function coefficients.

Abstract

We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.