Comultiplication rules for the double Schur functions and Cauchy identities

Understand

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.