Skew odd orthogonal characters and interpolating Schur polynomials

TL;DR

This paper introduces vertex operators for skew odd orthogonal characters, deriving identities and interpolating symmetric polynomials among classical groups, providing new proofs and explicit transition formulas.

Contribution

It develops new vertex operators for skew odd orthogonal characters and constructs interpolating symmetric polynomials among classical groups, with explicit transition formulas.

Findings

Derived Cauchy identity for skew odd orthogonal characters.

Provided new proofs of Jacobi--Trudi and Gelfand--Tsetlin identities.

Constructed three families of interpolating symmetric polynomials.

Abstract

We introduce two vertex operators to realize skew odd orthogonal characters $so_{\lambda/\mu}(x^{\pm})$ 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 $so_{\lambda/\mu}(x^{\pm})$. 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 $SO_{2n+1}(\mathbb{C})$, $SO_{2n}(\mathbb{C})$ and $Sp_{2n}(\mathbb{C})$, Their transition formulas are also explicitly given among symplectic and/or orthogonal characters and odd orthogonal characters.