On symmetric representations of

We introduce the notions of symmetric and symmetrizable representations $\text{SL}_2(\mathbb{Z})$. The linear $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are have congruence kernel. Conversely, one may also reconstruct data finite-dimensional symmetric, By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry some Weil at prime power levels, we prove that all symmetrizable. provide examples unsymmetrizable noncongruence subrepresentations one.