Comultiplication rules for the double Schur functions and Cauchy identities

Details Comultiplication rules for the double Schur functions and Cauchy identities Comultiplication rules for the double Schur functions and Cauchy identities

2008-07-14

arxiv.org/abs/0807.2127v3

Electronic Jour. Combinatorics 16 (2009), R13, 44pp

Mathematics

Combinatorics

44 pages, some corrections are made in sections 2.3 and 5.1

Scientific paper

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.

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