Lattice Paths and the Flagged Cauchy Determinant

We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions. By choosing different origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in terms of the complete symmetric functions in the full variable sets. The algebraic proof of this equivalence involves the Cauchy-Binet formula and mutli-Schur functions based on the complete super symmetric function. We also present an evaluation of the Cauchy determinant by the Jacobi symmetrizer.