Affine and cyclotomic $q$-Schur categories via webs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910910910562304 |
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| author | Shen, Yaolong Song, Linliang Wang, Weiqiang |
| author_facet | Shen, Yaolong Song, Linliang Wang, Weiqiang |
| contents | We formulate two new $\mathbb Z[q,q^{-1}]$-linear diagrammatic monoidal categories, the affine $q$-web category and the affine $q$-Schur category, as well as their respective cyclotomic quotient categories. Diagrammatic integral bases for the Hom-spaces of all these categories are established. In addition, we establish the following isomorphisms, providing diagrammatic presentations of these $q$-Schur algebras for the first time: (i)~ the path algebras of the affine $q$-web category to R.~Green's affine $q$-Schur algebras, (ii)~ the path algebras of the affine $q$-Schur category to Maksimau-Stroppel's higher level affine $q$-Schur algebras, and most significantly, (iii)~ the path algebras of the cyclotomic $q$-Schur categories to Dipper-James-Mathas' cyclotomic $q$-Schur algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine and cyclotomic $q$-Schur categories via webs Shen, Yaolong Song, Linliang Wang, Weiqiang Representation Theory Quantum Algebra We formulate two new $\mathbb Z[q,q^{-1}]$-linear diagrammatic monoidal categories, the affine $q$-web category and the affine $q$-Schur category, as well as their respective cyclotomic quotient categories. Diagrammatic integral bases for the Hom-spaces of all these categories are established. In addition, we establish the following isomorphisms, providing diagrammatic presentations of these $q$-Schur algebras for the first time: (i)~ the path algebras of the affine $q$-web category to R.~Green's affine $q$-Schur algebras, (ii)~ the path algebras of the affine $q$-Schur category to Maksimau-Stroppel's higher level affine $q$-Schur algebras, and most significantly, (iii)~ the path algebras of the cyclotomic $q$-Schur categories to Dipper-James-Mathas' cyclotomic $q$-Schur algebras. |
| title | Affine and cyclotomic $q$-Schur categories via webs |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2504.10270 |