Annals of Combinatorics 4 (2000) 285-297


Differential Symmetric Functions

Joseph P.S. Kung

Department of Mathematics, University of North Texas, Denton, Texas 76203, USA
kung@unt.edu

Received April 5, 1999

AMS Subject Classification: 05E05, 12H05, 15A15

Abstract. A (generalized) Wronskian is a determinant of the form 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.

Keywords: Wronskians, alternants, symmetric functions, Schur functions, Jacobi-Trudi identity


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