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Báo cáo toán học: "Symmetric functions for the generating matrix of the Yangian of gln(C)"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Symmetric functions for the generating matrix of the Yangian of gln(C). | Symmetric functions for the generating matrix of the Yangian of gln C Natasha Rozhkovskaya Department of Mathematics Kansas State University USA Submitted June 9 2009 Accepted Oct 13 2009 Published Oct 26 2009 Mathematics Subject Classification 05E05 05E10 17B37 Abstract Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in the Yangian are proved. As a corollary similar relations are deduced for shifted Schur polynomials. Introduction In this note we prove some combinatorial relations between the analogues of symmetric functions for the Yangian of the Lie algebra gln C . The applications of the results are illustrated by deducing properties of Capelli polynomials and shifted symmetric polynomials. Some of these properties were obtained for example in 16 from the definitions of shifted symmetric functions. Here due to the existence of evaluation homomorphism they become immediate consequences of similar combinatorial formulas in the Yangian. The elementary symmetric functions in the Yangian of the Lie algebra gln C are known to be generators of Bethe subalgebra. Bethe subalgebra finds numerous applications in quantum integrable models of XXX type and Gaudin type 10 11 12 . We describe the inverse of the universal differential operator for higher transfer matrices of XXX model. The author is very grateful to E. Mukhin for encouraging discussions and valuable remarks. She is grateful to the referee for the suggested improvements of the paper to P. Pyatov A. Chervov for sharing comments on the subject. The hospitality of Institut des Hautes Etudes Scientifiques and of Max Plank Institute for Mathematics in Bonn provided very inspiring atmosphere for the research. The project is supported in part by KSU Mentoring fellowship for WMSE. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R128 1 Notations and Preliminary facts The following notations will be used through the paper. All non-commutative .