Cumulants of Jack symmetric functions and $b$-conjecture

TL;DR

This paper proves a partial version of the $b$-conjecture by showing that certain coefficients related to Jack symmetric functions are polynomials in  with rational coefficients, advancing understanding of hypermap generating series.

Contribution

The paper demonstrates that coefficients of a multivariate generating series are polynomials in  with rational coefficients, providing a partial proof of the $b$-conjecture and revealing a key factorization property of Jack polynomials.

Findings

Coefficients are polynomials in  with rational coefficients.

A strong factorization property of Jack polynomials as  approaches 0.

Partial proof of the $b$-conjecture.

Abstract

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(x, y, z; t, 1+\beta)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $\psi(x, y, z; t, 1+\beta)$ in the power-sum basis are polynomials in $\beta$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $\beta$). We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $\psi(x, y, z; t, 1+ \beta)$ are polynomials in $\beta$ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter $\alpha$ tends to $0$, that may be of independent interest.