Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters

Abstract

We give bijective proofs for Jacobi–Trudi-type and Giambelli-type identities for symplectic and orthogonal characters. These proofs base on interpreting King and El-Sharkaway's symplectic tableaux, Proctor's odd and intermediate symplectic tableaux, Proctor's and King and Welsh's orthogonal tableaux, and Sundaram's odd orthogonal tableaux in terms of certain families of nonintersecting lattice paths. This work is intended to be the counterpart of the Gessel–Viennot proof of the Jacobi–Trudi identities for Schur functions for the case of symplectic and orthogonal characters.