Skew odd orthogonal characters and interpolating Schur polynomials
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914086425460736 |
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| author | Jing, Naihuan Li, Zhijun Wang, Danxia Ye, Chang |
| author_facet | Jing, Naihuan Li, Zhijun Wang, Danxia Ye, Chang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_15586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew odd orthogonal characters and interpolating Schur polynomials Jing, Naihuan Li, Zhijun Wang, Danxia Ye, Chang Representation Theory Combinatorics Quantum Algebra Primary: 05E05, Secondary: 17B37 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. |
| title | Skew odd orthogonal characters and interpolating Schur polynomials |
| topic | Representation Theory Combinatorics Quantum Algebra Primary: 05E05, Secondary: 17B37 |
| url | https://arxiv.org/abs/2502.15586 |