The $q$-Schur category and polynomial tilting modules for quantum $GL_n$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913860221403136 |
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| author | Brundan, Jonathan |
| author_facet | Brundan, Jonathan |
| contents | The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum $GL_n$ for all $n \geq 0$. Then we use Donkin's work on Ringel duality for $q$-Schur algebras to make precise the relationship between the $q$-Schur category and an integral form for the $U_q\mathfrak{gl}_n$-web category of Cautis, Kamnitzer and Morrison. We construct explicit integral bases for morphism spaces in the latter category, and extend the Cautis-Kamnitzer-Morrison theorem to polynomial representations of quantum $GL_n$ at a root of unity over a field of any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $q$-Schur category and polynomial tilting modules for quantum $GL_n$ Brundan, Jonathan Quantum Algebra Representation Theory 18B10 The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum $GL_n$ for all $n \geq 0$. Then we use Donkin's work on Ringel duality for $q$-Schur algebras to make precise the relationship between the $q$-Schur category and an integral form for the $U_q\mathfrak{gl}_n$-web category of Cautis, Kamnitzer and Morrison. We construct explicit integral bases for morphism spaces in the latter category, and extend the Cautis-Kamnitzer-Morrison theorem to polynomial representations of quantum $GL_n$ at a root of unity over a field of any characteristic. |
| title | The $q$-Schur category and polynomial tilting modules for quantum $GL_n$ |
| topic | Quantum Algebra Representation Theory 18B10 |
| url | https://arxiv.org/abs/2407.07228 |