Affine and cyclotomic Schur categories
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909439746899968 |
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| author | Song, Linliang Wang, Weiqiang |
| author_facet | Song, Linliang Wang, Weiqiang |
| contents | Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur category, for any commutative ring $\Bbbk$. We then formulate diagrammatic categories, the cyclotomic Schur categories, with arbitrary parameters at positive integral levels. Integral bases consisting of elementary diagrams are obtained for affine and cyclotomic Schur categories. A second diagrammatic basis, called a double SST basis, for any such cyclotomic Schur category is also established, leading to a conjectural higher level RSK correspondence. We show that the endomorphism algebras with the double SST bases are isomorphic to degenerate cyclotomic Schur algebras with their cellular bases, providing a first diagrammatic presentation of the latter. The presentations for the affine and cyclotomic Schur categories are much simplified when $\Bbbk$ is a field of characteristic zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Affine and cyclotomic Schur categories Song, Linliang Wang, Weiqiang Representation Theory Quantum Algebra Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur category, for any commutative ring $\Bbbk$. We then formulate diagrammatic categories, the cyclotomic Schur categories, with arbitrary parameters at positive integral levels. Integral bases consisting of elementary diagrams are obtained for affine and cyclotomic Schur categories. A second diagrammatic basis, called a double SST basis, for any such cyclotomic Schur category is also established, leading to a conjectural higher level RSK correspondence. We show that the endomorphism algebras with the double SST bases are isomorphic to degenerate cyclotomic Schur algebras with their cellular bases, providing a first diagrammatic presentation of the latter. The presentations for the affine and cyclotomic Schur categories are much simplified when $\Bbbk$ is a field of characteristic zero. |
| title | Affine and cyclotomic Schur categories |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2407.10119 |