Content evaluation and class symmetric functions

In this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over ℚ(n) that we call the basis of class symmetric functions. By definition this basis provides an algebra isomorphism between Λ and the Farahat-Higman algebra FH governing for all n the products of conjugacy classes in the center Z_n of the group algebra of the symmetric group S_n. We thus obtain a calculus of all connexion coefficients of Z_n inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients. We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras.