On symmetric representations of 𝑆𝐿₂(ℤ)

Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel

doi:10.1090/proc/16205

On symmetric representations of $\mathrm {SL}_2(\mathbb {Z})$
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by Siu-Hung Ng, Yilong Wang and Samuel Wilson;

Proc. Amer. Math. Soc. 151 (2023), 1415-1431

DOI: https://doi.org/10.1090/proc/16205

Published electronically: January 26, 2023

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Abstract:

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

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On symmetric representations of $\mathrm {SL}_2(\mathbb {Z})$
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Abstract: