On symmetric representations of 𝑆𝐿₂(ℤ)

Abstract

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.