Skew odd orthogonal characters and interpolating Schur polynomials

Abstract

We introduce two vertex operators to realize skew odd orthogonal characters $s{o}_{\hat{I}»/\hat{I}¼}(x^{\pm })$ and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method also gives new proofs of the Jacobi--Trudi identity and Gelfand--Tsetlin patterns for $s{o}_{\hat{I}»/\hat{I}¼}(x^{\pm })$. Moreover, combining the vertex operators related to characters of types $C,D$ (\cite{Ba1996,JN2015}) and the new vertex operators related to $B$-type characters, we obtain three families of symmetric polynomials that interpolate among characters of $S{O}_{2n+1}(\mathbb{C})$, $S{O}_{2n}(\mathbb{C})$ and $S{p}_{2n}(\mathbb{C})$, Their transition formulas are also explicitly given among symplectic and/or orthogonal characters and odd orthogonal characters.

Appendix with Xinyu Pan; 18pp