### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 041, 23 pages      arXiv:2208.05526      https://doi.org/10.3842/SIGMA.2024.041

### Skew Symplectic and Orthogonal Schur Functions

Naihuan Jing a, Zhijun Li b and Danxia Wang b
a) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
b) School of Science, Huzhou University, Huzhou, Zhejiang 313000, P.R. China

Received August 28, 2023, in final form May 12, 2024; Published online May 21, 2024

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.

Key words: skew orthogonal/symplectic Schur functions; Jacobi-Trudi identity; Gelfand-Tsetlin patterns; vertex operators.

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