On Vertex Operator Realization Of Symplectic Schur Functions

The character formulas of irreducible representations of the classical groups werefrst discovered in1925by Weyl using integral on the group manifold. Schur functionswere constructed as Bernstein operators by Bernstein[32]in the early days. Koike Kand Terada I[17]introduced the universal character of sp(2n) and expanded it as adeterminant form in1987, they also gave the corresponding function, namely, SymplecticSchur functions. Jing N[11,12]studied the Schur function, Schur-Q function and Hall-Littlewood function using vertex operators method in1991. Baker T H[2]gave the vertexoperator construction of the Symplectic and Orthogonal Schur functions in1996. Jing Nrealized the Schur function which is given by the Weyl character formula of type A byconstructing a vertex operator in1999. This paper improves Baker’s method and gives anew proof of the determinant expansion of Weyl character formula of type C.This article goes in this way: frst, it reviews of the research status of theory relatedto the Symplectic Schur functions; then it respectively introduces the basic results abouttype A and type C: the Weyl character formula and its determinantal expansion andknowledge of its structure constants; the last is the main part of this paper, it describesconstruction about the vertex operator of type A and the vertex operator realization ofSchur function, the construction about the vertex operator of type C and its properties,the vertex operator realization of Symplectic Schur functions.