On symmetric representations of 𝑆𝐿₂(ℤ)
| Autor: | Siu-Hung Ng, Yilong Wang, Samuel Wilson |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/16205 |
| Popis: | We introduce the notions of symmetric and symmetrizable representations of SL 2 ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . The linear representations of SL 2 ( Z ) {\operatorname {SL}_2(\mathbb {Z})} arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of SL 2 ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . By investigating a Z / 2 Z \mathbb {Z}/2\mathbb {Z} -symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of SL 2 ( Z ) {\operatorname {SL}_2(\mathbb {Z})} are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of SL 2 ( Z ) {\operatorname {SL}_2(\mathbb {Z})} that are subrepresentations of a symmetric one. |
| Databáze: | OpenAIRE |
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