Cumulants of Jack symmetric functions and $b$-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.

27 pages, 2 figures, to appear in Trans. Amer. Math. Soc