On cluster structures of bosonic extensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918515115556864 |
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| author | Bi, Yingjin |
| author_facet | Bi, Yingjin |
| contents | We study quantum cluster structures on bosonic extensions of quantum unipotent coordinate rings. For a positive braid group element $b\in \operatorname{Br}^+$, Kashiwara--Kim--Oh--Park introduced a subalgebra $\widehat{\mathcal A}(b)$ and conjectured that it admits a quantum cluster algebra structure whose cluster monomials belong to the global basis.
In this paper, we analyze Lusztig parametrizations of the global basis of $\widehat{\mathcal A}(b)$ and study their transition maps under braid moves. We prove that the resulting quantum cluster structure is independent of the chosen expression of $b$. Combining these ingredients, we prove the Kashiwara--Kim--Oh--Park conjecture for every \(b\in\operatorname{Br}^+\) in type ADE. Our proof is based on the compatibility between Lusztig parametrizations, braid moves, and cluster mutations, and is different from the approaches of Qin and of Kashiwara--Kim--Oh--Park. We also establish quantum \(T\)-system relations for generalized quantum minors and show that these minors occur as cluster variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cluster structures of bosonic extensions Bi, Yingjin Representation Theory 13F60, 17B37, 17B67 We study quantum cluster structures on bosonic extensions of quantum unipotent coordinate rings. For a positive braid group element $b\in \operatorname{Br}^+$, Kashiwara--Kim--Oh--Park introduced a subalgebra $\widehat{\mathcal A}(b)$ and conjectured that it admits a quantum cluster algebra structure whose cluster monomials belong to the global basis. In this paper, we analyze Lusztig parametrizations of the global basis of $\widehat{\mathcal A}(b)$ and study their transition maps under braid moves. We prove that the resulting quantum cluster structure is independent of the chosen expression of $b$. Combining these ingredients, we prove the Kashiwara--Kim--Oh--Park conjecture for every \(b\in\operatorname{Br}^+\) in type ADE. Our proof is based on the compatibility between Lusztig parametrizations, braid moves, and cluster mutations, and is different from the approaches of Qin and of Kashiwara--Kim--Oh--Park. We also establish quantum \(T\)-system relations for generalized quantum minors and show that these minors occur as cluster variables. |
| title | On cluster structures of bosonic extensions |
| topic | Representation Theory 13F60, 17B37, 17B67 |
| url | https://arxiv.org/abs/2506.00882 |