PreprintComultiplication rules for the double Schur functions and Cauchy identitiesA. I. MolevAbstractThe double Schur functions form a distinguished basis of the ring Lambda(xa) which is a multiparameter generalization of the ring of symmetric functions Lambda(x). The canonical comultiplication on Lambda(x) is extended to Lambda(xa) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual LittlewoodRichardson 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 LittlewoodRichardson coefficients provide a multiplication rule for the dual Schur functions. This paper is available as a pdf (352kB) file.
