Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Cumulants of Jack symmetric functions and b-conjecture (extended abstract)
Maciej Dolega  1, 2  , Valentin Féray  3  
1 : Faculty of Mathematics and Computer Science [Poznan]
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland -  Pologne
2 : Institute of Mathematics  (WROCLAW)  -  Website
Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland -  Pologne
3 : Institut für Mathematik [Zürich]  -  Website
Winterthurerstrasse 190, CH-8057 Zürich -  Suisse

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. 

  • Poster
Online user: 1