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Báo cáo toán học: "Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Pl¨cker u relations"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Pl¨cker u relations. | Bijective proofs for Schur function identities which imply Dodgson s condensation formula and Plucker relations Markus Fulmek Institut fur Mathematik der Universitat Wien Strudlhofgasse 4 A-1090 Wien Austria Markus.Fulmek@Univie.Ac.At Michael Kleber Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge MA 02139 USA Kleber@Math.Mit.Edu Submitted July 3 2000 Accepted March 7 2001. MR Subject Classifications 05E05 05E15 Abstract We present a method for bijective proofs for determinant identities which is based on translating determinants to Schur functions by the Jacobi-Trudi identity. We illustrate this method by generalizing a bijective construction which was first used by Goulden to a class of Schur function identities from which we shall obtain bijective proofs for Dodgson s condensation formula Plucker relations and a recent identity of the second author. 1 Introduction Usually bijective proofs of determinant identities involve the following steps cf. e.g 19 Chapter 4 or 23 24 Expansion of the determinant as sum over the symmetric group Interpretation of this sum as the generating function of some set of combinatorial objects which are equipped with some signed weight Construction of an explicit weight- and sign-preserving bijection between the respective combinatorial objects maybe supported by the construction of a signreversing involution for certain objects. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R16 1 Here we will present another method of bijective proofs for determinant identitities which involves the following steps First we replace the entries ai j of the determinants by h i-i j where hm denotes the m-th complete homogeneous function Second by the Jacobi-Trudi identity we transform the original determinant identity into an equivalent identity for Schur functions Third we obtain a bijective proof for this equivalent identity by using the interpretation of Schur functions in terms of nonintersecting lattice paths. In this paper we