Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

Maciej Dolega,Valentin Féray

doi:10.46298/dmtcs.6322

Abstract

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

Highlights

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series
A key step in our proof of Theorem 1.2 is a strong factorization of Jack polynomials when α tends to zero
For r = 2, the theorem asserts that Jλ(α1⊕) λ2 = Jλ(α1 ) Jλ(α2 ) (1 + O(α)), i.e. that Jλ(α1⊕) λ2 factorizes when α tends to 0. This follows from the explicit expression of Jack polynomials for α = 0 [15, Proposition 7.6]

Summary

Introduction

Let Jλ(α)(x) be the Jack symmetric function indexed by a partition λ in the infinite alphabet x. In the case α = 1, the quantity hτμ,ν(0) enumerates connected hypergraphs embedded into oriented surfaces with vertex-, edge- and face-degree distributions given by μ, ν and τ . M where the summation index runs over all rooted hypermaps M vertex-, edge- and face-degree distributions given by μ, ν and τ , and η(M) is a nonnegative integer equals to 0 if and only if M is orientable. The nonnegativity and the integrality of the coefficients seem out of reach with our approach

Strong factorization of Jack polynomials