Cumulants of Jack symmetric functions and the 𝑏-conjecture

Dołęga, Maciej; Féray, Valentin

doi:10.1090/tran/7191

Cumulants of Jack symmetric functions and the $b$-conjecture
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Trans. Amer. Math. Soc. 369 (2017), 9015-9039 Request permission

Abstract:

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+\beta )$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $\psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+\beta )$ in the power-sum basis are polynomials in $\beta$ with non-negative integer coefficients (by construction, these coefficients are rational functions in $\beta$).

We partially prove this conjecture, nowadays called the $b$-conjecture, by showing that coefficients of $\psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+ \beta )$ 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 $0$, which may be of independent interest.

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