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Title: On symmetric representations of 𝑆𝐿₂(β„€)

We introduce the notions of symmetric and symmetrizable representations ofSL2⁑<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}. The linear representations ofSL2⁑<#comment/>(Z){\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 ofSL2⁑<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}. By investigating aZ/2Z\mathbb {Z}/2\mathbb {Z}-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations ofSL2⁑<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations ofSL2⁑<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}that are subrepresentations of a symmetric one.

 
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Award ID(s):
1664418
PAR ID:
10476836
Author(s) / Creator(s):
; ;
Publisher / Repository:
American Mathematical Society
Date Published:
Journal Name:
Proceedings of the American Mathematical Society
ISSN:
0002-9939
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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