Universal symplectic/orthogonal functions and general branching rules

TL;DR

This paper introduces universal symplectic and orthogonal functions, realizes them via vertex operators, and establishes their branching rules, transition formulas, and connections to interpolating Schur functions, broadening the understanding of classical and generalized characters.

Contribution

It constructs universal symplectic and orthogonal functions with vertex operator realizations, deriving their branching rules and transition formulas, and links interpolating Schur functions to orthosymplectic Schur polynomials.

Findings

Universal symplectic functions include various classical symplectic characters.

Universal orthogonal functions generalize multiple orthogonal characters.

Interpolating Schur functions are shown to equal orthosymplectic Schur polynomials.

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).