ON SYMMETRIC REPRESENTATIONS OF SL2(ℤ)

Document Type

Article

Publication Date

4-1-2023

Abstract

We introduce the notions of symmetric and symmetrizable representations of SL2(ℤ). The linear representations of SL2(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 SL2(ℤ). By investigating a ℤ/2ℤ-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of SL2(ℤ) are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of SL2(ℤ) that are subrepresentations of a symmetric one.

Publication Source (Journal or Book title)

Proceedings of the American Mathematical Society

First Page

1415

Last Page

1431

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