On symmetric representations of \(\mathrm{SL}_2(\mathbb{Z})\). (English) Zbl 1521.11027
Summary: We introduce the notions of symmetric and symmetrizable representations of \(\mathrm{SL}_2(\mathbb{Z})\). The linear representations of \(\mathrm{SL}_2(\mathbb{Z})\) 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 \(\mathrm{SL}_2(\mathbb{Z})\). By investigating a \(\mathbb{Z}/2\mathbb{Z}\)-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of \(\mathrm{SL}_2(\mathbb{Z})\) are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of \(\mathrm{SL}_2(\mathbb{Z})\) that are subrepresentations of a symmetric one.
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
| 11E08 | Quadratic forms over local rings and fields |
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 20C35 | Applications of group representations to physics and other areas of science |
| 20G05 | Representation theory for linear algebraic groups |
Software:
SL2RepsReferences:
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