Lattice Paths And Symmetric Functions

The diagonalization of the Cauchy kernel is one of the most important theorems inthe theory of symmetric functions. This theorem is known as the Cauchy theorem.The aim of this work is to give a lattice path approach to the Cauchy theorem onSchur functions, and to present lattice path interpretations of (isobaric) divided di?er-ences and special Schubert polynomials. The latter one is our e?ort in attacking thediagonalization of the Cauchy kernel on Schubert polynomials (the noncommutativeversion conjectured by Alain Lascoux is still open).We start by obtaining a ?agged form of the Cauchy determinant and establisha correspondence between this determinant and nonintersecting lattice paths, fromwhich it follows that Cauchy identity on Schur functions. By choosing di?erent originsand destinations for the lattice paths, we are led to an identity of Gessel on theCauchy sum of Schur functions in terms of the complete symmetric functions in thefull variable sets. The algebraic proof of this equivalence involves the Cauchy-Binetformula and multi-Schur functions based on the complete super symmetric function.We also present an evaluation of the Cauchy determinant by the Jacobi symmetrizer.Then we obtain a tableau definition of the skew Schubert polynomials named byLascoux, which are defined as ?agged double skew Schur functions. These polynomi-als are in fact Schubert polynomials in two sets of variables indexed by 321-avoidingpermutations. From the divided di?erence definition of the skew Schubert polyno-mials, we construct a lattice path interpretation based on the Chen-Li-Louck pairinglemma. The lattice path explanation immediately leads to the determinantal defini-tion and the tableau definition of the skew Schubert polynomials. For the case of asingle variable set, the skew Schubert polynomials reduce to ?agged skew Schur func-tions as studied by Wachs and by Billey, Jockusch, and Stanley. Moreover, we presentan expression for the ?agged Schur function in terms of isobaric operators acting ona monomial.Finally, we find lattice path interpretations for the Giambelli identity and theLascoux-Pragacz identity for super-Schur functions. For the super-Lascoux-Pragaczidentity, the lattice path construction is related to the code of the partition whichdetermines the directions of the lines parallel to the y-axis in the lattice.