On symmetric representations of ${\text{SL}}_2(\mathbb{Z})$

Abstract

We introduce the notions of symmetric and symmetrizable representations of ${\text{SL}}_2(\mathbb{Z})$. The linear representations of ${\text{SL}}_2(\mathbb{Z})$ 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 ${\text{SL}}_2(\mathbb{Z})$. 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 ${\text{SL}}_2(\mathbb{Z})$ are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of ${\text{SL}}_2(\mathbb{Z})$ that are subrepresentations of a symmetric one.

14 pages

Cited by in corpus (3)

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