Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909322390274048 |
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| author | Gruber, Jonathan |
| author_facet | Gruber, Jonathan |
| contents | Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ of finite-dimensional $\mathbf{G}$-modules and study the associated quotient category $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient. Our main results are a 'linkage principle' and a 'translation principle' for tensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essential image in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of $\mathrm{Rep}(\mathbf{G})$. We first show that $\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in $\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidal structure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a large extent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$. These results can be combined to give an external tensor product decomposition $\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes \mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$ denotes the Verlinde category of $\mathbf{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07796 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups Gruber, Jonathan Representation Theory 20G05 (primary), 20G42, 17B10, 17B55 (secondary) Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ of finite-dimensional $\mathbf{G}$-modules and study the associated quotient category $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient. Our main results are a 'linkage principle' and a 'translation principle' for tensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essential image in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of $\mathrm{Rep}(\mathbf{G})$. We first show that $\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in $\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidal structure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a large extent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$. These results can be combined to give an external tensor product decomposition $\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes \mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$ denotes the Verlinde category of $\mathbf{G}$. |
| title | Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups |
| topic | Representation Theory 20G05 (primary), 20G42, 17B10, 17B55 (secondary) |
| url | https://arxiv.org/abs/2304.07796 |