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


Authors: Maciej Dołęga and Valentin Féray
Journal: Trans. Amer. Math. Soc. 369 (2017), 9015-9039
MSC (2010): Primary 05E05
DOI: https://doi.org/10.1090/tran/7191
Published electronically: September 7, 2017
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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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Additional Information

Maciej Dołęga
Affiliation: Wydział Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza, Collegium Mathematicum, Umultowska 87, 61-614 Poznań, Poland – and – Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: maciej.dolega@amu.edu.pl

Valentin Féray
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Email: valentin.feray@math.uzh.ch

DOI: https://doi.org/10.1090/tran/7191
Keywords: Jack symmetric functions, cumulants, Laplace-Beltrami operator
Received by editor(s): January 16, 2016
Received by editor(s) in revised form: January 23, 2017
Published electronically: September 7, 2017
Additional Notes: The first author was supported by Agence Nationale de la Recherche, grant ANR 12-JS02-001-01 “Cartaplus” and by NCN, grant UMO-2015/16/S/ST1/00420. The second author was partially supported by grant SNF-149461 “Dual combinatorics of Jack polynomials”.
Article copyright: © Copyright 2017 American Mathematical Society