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Communication Dans Un Congrès Discrete Mathematics and Theoretical Computer Science Année : 2020

Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

Résumé

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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Dates et versions

hal-02173382 , version 1 (04-07-2019)

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Maciej Dolega, Valentin Féray. Cumulants of Jack symmetric functions and b-conjecture (extended abstract). 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6322⟩. ⟨hal-02173382⟩
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