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

TL;DR

This paper explores symmetric and symmetrizable representations of SL_2(Z), establishing their properties, connections to modular tensor categories, and demonstrating that all finite-dimensional congruence representations are symmetrizable, with examples of noncongruence cases.

Contribution

It introduces the concepts of symmetric and symmetrizable representations of SL_2(Z), proves all finite-dimensional congruence representations are symmetrizable, and provides examples of noncongruence representations.

Findings

All finite-dimensional congruence representations are symmetrizable.

Symmetric representations have congruence kernels.

Examples of noncongruence, unsymmetrizable representations are provided.

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.