Cumulants of Jack symmetric functions and the $b$-conjecture


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 the Jack-deformation parameter $\alpha$ tends to $0$, that may be of independent interest.

Trans. Amer. Math. Soc., 369 (12), 9015–9039, 2017