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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Schur functions and alternating sums. | Schur functions and alternating sums Marc A. A. van Leeuwen Universite de Poitiers Departement de Mathematiques UFR Sciences SP2MI Teleport 2 BP 30179 86962 Futuroscope Chasseneuil Cedex France Marc.van-Leeuwen@math.univ-poitiers.fr http www-math.univ-poitiers.fr maavl Dedicated to Richard Stanley on the occasion of his 60th birthday Submitted Apr 18 2005 Accepted Feb 13 2006 Published Feb 22 2006 Mathematics Subject Classifications 05E05 05E10 Abstract We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations notably we shall discuss the Pieri and Murnaghan-Nakayama rules the Jacobi-Trudi identity and its dual Von Nagelsbach-Kostka identity their proofs using the correspondence with lattice paths of Gessel and Viennot and finally the Littlewood-Richardson rule. Our our goal is to show that the mentioned statements are closely related and can be proved using variations of the same basic technique. We also want to emphasise the central part that is played by matrices over 0 1 and over N we show that the Littlewood-Richardson rule as generalised by Zelevinsky has elegant formulations using either type of matrix and that in both cases it can be obtained by two successive reductions from a large signed enumeration of such matrices where the sign depends only on the row and column sums of the matrix. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2006 A5 1 0 Introduction 0. Introduction. Many of the more interesting combinatorial results and correspondences in the basic theory of symmetric functions involve Schur functions or more or less equivalently the notions of semistandard tableaux or horizontal strips. Yet the introduction of these notions in any of the many possible ways is not very natural when considering only symmetric functions cf. Stan 7.10 . One way the importance of Schur functions can be motivated is by representation theory interpreting symmetric functions in the representation theory .
