Schur Functions and Alternating Sums

  • Marc A. A. van Leeuwen


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 Nägelsbach-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 $\Bbb N$; 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.