Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Comultiplication rules for the double Schur functions and Cauchy identities. | Comultiplication rules for the double Schur functions and Cauchy identities A. I. Molev School of Mathematics and Statistics University of Sydney NSW 2006 Australia alexm@. au Submitted Aug 27 2008 Accepted Jan 17 2009 Published Jan 23 2009 Mathematics Subject Classifications 05E05 Abstract The double Schur functions form a distinguished basis of the ring A x a which is a multiparameter generalization of the ring of symmetric functions A x . The canonical comultiplication on A x is extended to A x 1 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. Contents 1 Introduction 2 2 Double and supersymmetric Schur functions 6 Definitions and preliminaries. 6 Analogues of classical bases . 9 Duality isomorphism. 10 Skew double Schur functions. 11 3 Cauchy identities and dual Schur functions 14 Definition of dual Schur functions and Cauchy identities . 14 Combinatorial presentation. 17 Jacobi-Trudi-type formulas . 20 Expansions in terms of Schur functions. 22 THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R13 1 4 Dual Littlewood-Richardson polynomials 29 5 Transition matrices 33 Pairing between the double and dual symmetric functions. 33 Kostka-type and character polynomials. 38 6 Interpolation formulas 40 Rational expressions for the transition coefficients. 40 Identities with dimensions of skew diagrams. 41 1 .
