Universal symplectic/orthogonal functions and general branching rules

Abstract

In this paper, we first introduce a family of universal symplectic functions $sp_\lambda(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_\lambda(\mathbf{x}^{\pm})$, odd symplectic characters $sp_\lambda(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_\lambda(\mathbf{z})$ and intermediate symplectic characters as subfamilies. We then realize the universal symplectic functions by vertex operators, which naturally lead to their skew versions, and show that $sp_\lambda(\mathbf{x}^{\pm};\mathbf{z})$ obey the general branching rules. This also gives the Gelfand-Tsetlin representations of odd symplectic characters and a transition formula between odd symplectic characters and symplectic Schur functions. Secondly we introduce a family of universal orthogonal functions $o_\lambda(\mathbf{x}^{\pm};\mathbf{z})$ and their skew versions in a similar manner, and we provide their vertex operator realizations and obtain transition formulas and the branching rule. The universal orthogonal functions $o_\lambda(\mathbf{x}^{\pm};\mathbf{z})$ generalize orthogonal Schur functions $o_\lambda(\mathbf{x}^{\pm})$, odd orthogonal Schur functions $so_\lambda(\mathbf{x}^{\pm})$, universal orthogonal characters $o_\lambda(\mathbf{z})$ as well as intermediate orthogonal characters. Thirdly, we give vertex operator realizations for the $CB$-interpolating Schur functions $s^{CB}_\lambda(x;\beta)$ introduced by Bisi and Zygouras (Adv. Math., 2022) and the $DB$-interpolating Schur functions $s^{DB}_\lambda(x;\beta)$ interpolating between characters of type $D$ and $B$. As an application, we show $s^{CB}_\lambda(x;\beta)$ are equal to the orthosymplectic Schur polynomials $spo_\lambda(x/\beta)$, thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723).