Annals of Combinatorics 4 (2000) 285-297Differential Symmetric Functions Joseph P.S. Kung Department of Mathematics, University of North Texas, Denton,
Texas 76203, USA 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" x_{1}, x_{2}, ..., x_{n} 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 References 1. A. Aitken, Note on dual symmetric functions, Proc. Edinburgh Math. Soc. 2 (1931) 164– 167. 2. W.Y.C. Chen and J.D. Louck, The factorial Schur function, J. Math. Phys. 34 (1993) 4144– 4160. 3. J.P.S. Kung and G.-C. Rota, On the differential invariants of a linear ordinary differential equation, Proc. Roy. Soc. Edinburgh 89A (1981) 111–123. 4. I.G. MacDonald, Symmetric functions and Hall polynomials, 2nd Ed., Oxford University Press, Oxford, 1995. 5. T. Muir, A treatise on the theory of determinants, W.H. Metzler, Ed., Longmans, London, 1933; Dover, New York, 1960, reprint. 6. S. Pincherle, Le Operazioni Distributive e le Loro Applicazioni All’analisi, Zanichelli, Bologna, 1901. 7. R.P. Stanley, Theory and applications of plane partitions, I and II, Stud. Appl. Math. 50 (1970) 168–188, 259–279. |