Comultiplication rules for the double Schur functions and Cauchy identities

Abstract

The double Schur functions form a distinguished basis of the ring \Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended to \Lambda(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood-Richardson coefficients provide a multiplication rule for the dual Schur functions.