Cumulants of Jack symmetric functions and $b$-conjecture

Maciej Dołęga, Valentin Féray

Published 2016-01-07Version 1

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 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 $\alpha$ tends to $0$, that may be of independent interest.