Doctor of Philosophy (PhD)
We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an action of the Galois group Gal(Qbar/Q) on their simple objects.
Wilson, Samuel Nathan, "SL(2,Z) Representations and 2-Semiregular Modular Categories" (2023). LSU Doctoral Dissertations. 6094.