The double Schur functions form a distinguished basis of the ring ( xjja) which is a multiparameter generalization of the ring of symmetric functions ( x). The canonical comultiplication on ( x) is extended to ( xjja) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood{Richardson coecien ts in two dieren t 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 coecien ts provide a multiplication rule for the dual Schur functions.
['A. I. Molev']
Comultiplication rules for the double Schur functions and Cauchy identities