Explicit monomial expansions of the generating series for connection coefficients

TL;DR

This paper provides explicit algebraic formulas for generating series of connection coefficients in symmetric group algebras, linking combinatorial structures with symmetric functions and zonal polynomials, with applications in random matrix theory.

Contribution

It introduces a new algebraic approach to derive simpler formulas for these generating series and provides explicit expressions for zonal polynomials of specific types.

Findings

New algebraic formulas for connection coefficient generating series

Simplified formulas for double coset algebra

Explicit expression for certain zonal polynomials

Abstract

This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces 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 derive a new explicit expression for zonal polynomials indexed by partitions of type [a,b,1^(n-a-b)].