Universal symplectic/orthogonal functions and general branching rules
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916499787087872 |
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| author | Jin, Zhihong Jing, Naihuan Li, Zhijun Wang, Danxia |
| author_facet | Jin, Zhihong Jing, Naihuan Li, Zhijun Wang, Danxia |
| contents | In this paper, we first introduce a family of universal symplectic functions $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_λ(\mathbf{x}^{\pm})$, odd symplectic characters $sp_λ(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_λ(\mathbf{z})$ 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 $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ 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 $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ 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 $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ generalize orthogonal Schur functions $o_λ(\mathbf{x}^{\pm})$, odd orthogonal Schur functions $so_λ(\mathbf{x}^{\pm})$, universal orthogonal characters $o_λ(\mathbf{z})$ as well as intermediate orthogonal characters.
Thirdly, we give vertex operator realizations for the $CB$-interpolating Schur functions $s^{CB}_λ(x;β)$ introduced by Bisi and Zygouras (Adv. Math., 2022) and the $DB$-interpolating Schur functions $s^{DB}_λ(x;β)$ interpolating between characters of type $D$ and $B$. As an application, we show $s^{CB}_λ(x;β)$ are equal to the orthosymplectic Schur polynomials $spo_λ(x/β)$, thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00767 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Universal symplectic/orthogonal functions and general branching rules Jin, Zhihong Jing, Naihuan Li, Zhijun Wang, Danxia Combinatorics Quantum Algebra Primary: 05E05, Secondary: 17B37 In this paper, we first introduce a family of universal symplectic functions $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_λ(\mathbf{x}^{\pm})$, odd symplectic characters $sp_λ(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_λ(\mathbf{z})$ 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 $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ 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 $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ 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 $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ generalize orthogonal Schur functions $o_λ(\mathbf{x}^{\pm})$, odd orthogonal Schur functions $so_λ(\mathbf{x}^{\pm})$, universal orthogonal characters $o_λ(\mathbf{z})$ as well as intermediate orthogonal characters. Thirdly, we give vertex operator realizations for the $CB$-interpolating Schur functions $s^{CB}_λ(x;β)$ introduced by Bisi and Zygouras (Adv. Math., 2022) and the $DB$-interpolating Schur functions $s^{DB}_λ(x;β)$ interpolating between characters of type $D$ and $B$. As an application, we show $s^{CB}_λ(x;β)$ are equal to the orthosymplectic Schur polynomials $spo_λ(x/β)$, thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723). |
| title | Universal symplectic/orthogonal functions and general branching rules |
| topic | Combinatorics Quantum Algebra Primary: 05E05, Secondary: 17B37 |
| url | https://arxiv.org/abs/2209.00767 |