The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910801539891200 |
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| author | Utiralova, Alexandra |
| author_facet | Utiralova, Alexandra |
| contents | Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups $GL(X)$ in the Verlinde category $Ver_p$ in characteristic $p$. The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup $B(X)\subset GL(X)$. We do this by reducing the general case to the study of representations of the group $GL(X)$ for $X=L_m\oplus L_{n}$ using the method of odd reflections. On the category of representations of $GL(L_m\oplus L_{n})$ we introduce the structure of the highest weight category, as well as the categorical action of $\widehat{\mathfrak{sl}}_p$ through translation functors. It allows us to understand projective and injective objects, BGG reciprocity, duality and lowest weights for simple modules, and standard filtration multiplicities for projective objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic Utiralova, Alexandra Representation Theory Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups $GL(X)$ in the Verlinde category $Ver_p$ in characteristic $p$. The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup $B(X)\subset GL(X)$. We do this by reducing the general case to the study of representations of the group $GL(X)$ for $X=L_m\oplus L_{n}$ using the method of odd reflections. On the category of representations of $GL(L_m\oplus L_{n})$ we introduce the structure of the highest weight category, as well as the categorical action of $\widehat{\mathfrak{sl}}_p$ through translation functors. It allows us to understand projective and injective objects, BGG reciprocity, duality and lowest weights for simple modules, and standard filtration multiplicities for projective objects. |
| title | The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2501.15778 |