Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Lattice path proofs of extended Bressoud-Wei and Koike skew Schur function identities. | Lattice path proofs of extended Bressoud-Wei and Koike skew Schur function identities A. M. Hamel Department of Physics and Computer Science Wilfrid Laurier University Waterloo Ontario N2L 3C5 Canada . Kingt School of Mathematics University of Southampton Southampton SO17 1BJ England Submitted Nov 29 2010 Accepted Feb 14 2011 Published Feb 21 2011 Mathematics Subject Classification 05E05 Abstract Our recent paper 5 provides extensions to two classical determinantal results of Bressoud and Wei and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs. Keywords Schur functions lattice paths 1 Introduction Our recent paper 5 provides proofs of certain generalizations of two classical determinan-tal identities one by Bressoud and Wei 1 and one by Koike 8 . Both of these identities are extensions of the Jacobi-Trudi identity an identity that provides a determinantal representation of the Schur function. Here we provide lattice path proofs of these generalized idetities. We give the barest of background details and notation referring the reader instead to our earlier paper 5 and to Macdonald 10 or Stanley 11 for general symmetric function background knowledge. e-mail ahamel@ le-mail THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P47 1 Let p be the set of all partitions including the zero partition. Recall that in Frobenius notation each partition A Al A2 . G P is written in the form A ai a a 1 b1 b2 br J with a1 a2 ar 0 and b1 b2 br 0 where ai Ak k and bk A k k for k 1 2 . r with A the partition conjugate to A. Here r r A the rank of A which is defined to be the maximum value of k such that Ak k. The partition A is said to have length A Al b1 1 and weight A A1 A2 a1 b1 a2 b2 ar br r. The case r 0 corresponds to the zero partition A 0 0 0 . of length A 0 and weight A 0. For any integer t let T _ i X _ 1 a2 ar _ _ for k 1 2 . r i . pt iA b1 12 br G P ak bk t r 01 r 2 I b1 b2 br i and r
