Skew odd orthogonal characters and interpolating Schur polynomials

TL;DR

This work extends the vertex-operator framework to skew odd orthogonal characters $so_{\lambda/\mu}(x^{\pm})$, establishing Jacobi–Trudi and GT-pattern descriptions and a Cauchy-type identity via Toeplitz-Hankel-type determinants. It then unifies the three classical families of Lie group characters by constructing three interpolating Schur polynomials $s^{BD}_\lambda(x;\alpha)$, $s^{BC}_\lambda(x;\alpha)$, $s^{CD}_\lambda(x;\alpha)$ that interpolate among $so_\lambda$, $o_\lambda$, and $sp_\lambda$ with explicit determinant formulas and transition relations. The results provide a cohesive operator-based route to symmetric-function identities across types $B$, $C$, and $D$, supply new proofs of known formulas, and reveal structured connections via GT patterns, Cauchy-type identities, and Toeplitz/Hankel determinants, with potential applications in representation theory and algebraic combinatorics.

Abstract

We introduce two vertex operators to realize skew odd orthogonal characters $so_{λ/μ}(x^{\pm})$ 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 $so_{λ/μ}(x^{\pm})$. 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.

Theorems & Definitions (35)

• Proposition 2.1
• proof
• Proposition 2.2
• Corollary 2.3
• Proposition 3.1
• proof
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• proof