Skew Symplectic and Orthogonal Schur Functions

TL;DR

This paper introduces new symmetric functions related to symplectic and orthogonal Schur functions using vertex operator representations, providing explicit formulas, identities, and combinatorial models that unify various types of skew Schur functions.

Contribution

It defines two families of symmetric functions as skew symplectic and orthogonal Schur polynomials via vertex operators, deriving identities and combinatorial structures that extend prior work.

Findings

Derived Jacobi-Trudi identities for these functions

Established Gelfand-Tsetlin pattern representations

Obtained Cauchy-type identities for the new functions

Abstract

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D.