On symmetric representations of $\text{SL}_2(\mathbb{Z})$
- Resource Type
- Working Paper
- Authors
- Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel
- Source
- Proceedings of the American Mathematical Society, Volume 151, Number 4, April 2023, Pages 1415--1431
- Subject
- Mathematics - Quantum Algebra
Mathematics - Category Theory
Mathematics - Group Theory
Mathematics - Number Theory
Mathematics - Representation Theory
- Language
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.
Comment: 14 pages