Crystal base of the negative half of quantum orthosymplectic superalgebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915583007653888 |
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| author | Jang, Il-Seung Kwon, Jae-Hoon Uruno, Akito |
| author_facet | Jang, Il-Seung Kwon, Jae-Hoon Uruno, Akito |
| contents | We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crystal base of the negative half of quantum orthosymplectic superalgebra Jang, Il-Seung Kwon, Jae-Hoon Uruno, Akito Quantum Algebra Representation Theory 17B37, 17B10 We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding. |
| title | Crystal base of the negative half of quantum orthosymplectic superalgebra |
| topic | Quantum Algebra Representation Theory 17B37, 17B10 |
| url | https://arxiv.org/abs/2510.24451 |