Representation of Symmetric Functions as Gram Determinants

Abstract

We introduce a representation of symmetric functions as determinants of Gram matrices on the space of rational functions in one variable. We use a bilinear map on the space of rational functions that allows us to identify rational functions with linear functionals. Using basic properties of Gram matrices and rational functions we obtain results about symmetric functions, like a general Jacobi–Trudi identity, and an expansion formula for general symmetric functions in terms of Schur functions. Our approach is suitable to study in a unified way the confluent “symmetric” (or pseudo-symmetric) functions proposed by Rota, and several kinds of generalized symmetric functions, like the factorial Schur functions and symmetric functions in several sets of indeterminates.