Cumulants of Jack symmetric functions and b-conjecture (extended abstract)
1 : Wydział Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza
Poznan -
Pologne
2 : Instytut Matematyczny, Uniwersytet Wrocawski
Wroclaw -
Pologne
3 : Institut fu ̈r Mathematik, Universita ̈t Zu ̈rich
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.
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