Differential Symmetric Functions

Abstract

A (generalized) Wronskian is a determinant of the form det\( [D^{\alpha_i} x_j]_{1 \leq i, j \leq n} \), where D is the differential operator. Wronskians are differential analogs of alternants. Imitating Jacobi's definition of Schur functions as quotients of two alternants, we define differential Schur functions as quotients of two Wronskians. Differential Schur functions are absolute invariants of the general linear group acting on the vector space spanned by the “functions”x1, x2,...,xn and they generate all such invariants. Thus, they provide the building blocks for a theory of differential symmetric functions. In this paper, we prove analogs of two basic identities: the Nagelsbach identity, which expressed Schur functions as determinants of elementary symmetric functions, and the Jacobi-Trudi identity, which expresses Schur functions as determinants of complete homogeneous symmetric functions. The proofs offer insight into the nature of these identities. In particular, the Jacobi-Trudi identity can be viewed as a change-of-basis formula.