Moments of normally distributed random matrices given by generating series for connection coefficients- Explicit algebraic computation

The class algebra and the double coset algebra are two classical commutative subalgebras of the group algebra of the symmetric group. The connexion coefficients of these two algebraic structures are important numbers with significant applications. From a combinatorial point of view, they give the number of factorizations of a given permutation into the ordered product of permutations with specific cyclic properties and count in some cases the number of hypermaps and constellations on (locally) orientable surfaces. They are also of notable interest in the study of Schur and zonal polynomials as well as in the theory of the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore as shown by Hanlon, Stanley, Stembridge (1992), the respective generating series of these coefficients in the basis of power sum symmetric functions are equal to the mathematical expectation of the trace of (XUYU∗)n where X and Y are given symmetric (respectively hermitian) matrices, U is a random real (respectively complex) valued square matrix of standard normal distribution and n a non negative integer.This paper is devoted to the explicit evaluation of these series in terms of monomial symmetric functions. Morales and Vassilieva (2009, 2011) and Vassilieva (2013) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type [a,b,1n−a−b].