Zonal polynomials via Stanley's coordinates and free cumulants

JOURNAL OF ALGEBRA

卷:334

Article

Feray, Valentin¹  Sniady, Piotr^2,3 

[1] Univ Bordeaux 1, LaBRI, F-33400 Talence, France

[2] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

[3] Univ Wroclaw, Inst Math, PL-50384 Wroclaw, Poland

关键词: Zonal polynomials;    Free cumulants;    Symmetric functions;    Algebraic combinatorics;    Maps on non-oriented surfaces;   

DOI  :  10.1016/j.jalgebra.2011.03.008

来源: Elsevier

【 摘 要 】

We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanley's multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of Lassalle for Jack polynomials in the special case of zonal polynomials. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free