2011/11/26 by Ekaterina A. Vassilieva, Vassilieva, Ekaterina A.
Mathematics · #05E05 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1111.6215
openalex publication_date 2011/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to the explicit computation of generating series for\nthe connection coefficients of two commutative subalgebras of the group algebra\nof the symmetric group, the class algebra and the double coset algebra. As\nshown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral\ndistribution of some random matrices that are of interest to statisticians.\nMorales and Vassilieva (2009, 2011) found explicit formulas for these\ngenerating series in terms of monomial symmetric functions by introducing a\nbijection between partitioned hypermaps on (locally) orientable surfaces and\nsome decorated forests and trees. Thanks to purely algebraic means, we recover\nthe formula for the class algebra and provide a new simpler formula for the\ndouble coset algebra. As a salient ingredient, we derive a new explicit\nexpression for zonal polynomials indexed by partitions of type [a,b,1^(n-a-b)].\n