Schur functions can be defined as determinants of the Jacobi--Trudi matrices. The Plucker relations among minors of matrices then give rise to a family of quadratic relations among Schur functions. These relations have a simple combinatorial realization in terms of Gessel--Viennot style nonintersecting lattice paths.

An observation about representations (of quantum affine algebras) gives rise to the odd-sounding problem of finding representations of the symplectic or orthogonal (type B/C/D) Lie algebras whose characters obey the Schur function (type A character) quadratic relations. We can prove that this question has a unique solution from the symmetric functions point of view, but the representations which solve it still need a natural construction.