Comultiplication rules for the double Schur functions and Cauchy identities

TL;DR

This paper extends the theory of double Schur functions by establishing comultiplication rules, calculating dual Littlewood-Richardson coefficients, and proving multiparameter Cauchy identities, thereby enriching the algebraic structure of symmetric functions.

Contribution

It introduces comultiplication rules for double Schur functions and develops multiparameter Cauchy identities, along with dual Schur functions and their properties.

Findings

Derived comultiplication rules for double Schur functions.

Calculated dual Littlewood-Richardson coefficients.

Proved multiparameter analogues of the Cauchy identity.

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.