Skew odd orthogonal characters and interpolating Schur polynomials

Abstract

We introduce two vertex operators to realize skew odd orthogonal characters $s{o}_{\lambda /\mu }\left(x^{\pm }\right)$ and derive the Cauchy identity for the skew characters via the Toeplitz–Hankel-type determinant as an analog of Schur functions. The method also gives new proofs of the Jacobi–Trudi identity and Gelfand–Tsetlin patterns for $s{o}_{\lambda /\mu }\left(x^{\pm }\right)$. Moreover, combining the vertex operators related to characters of types $C,D$ (Baker, J. Phys. A. 29(12) (1996), 3099–3117; Jing and Nie, Ann. Combin. 19 (2015), 427–442) and the new vertex operators related to $B$-type characters, we obtain three families of symmetric polynomials that interpolate among characters of ${\mathrm{SO}}_{2n+1}\left(C\right)$, ${\mathrm{SO}}_{2n}\left(C\right)$, and ${\mathrm{Sp}}_{2n}\left(C\right)$. Their transition formulae are also explicitly given among symplectic, orthogonal, and odd orthogonal characters.