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Indicators of Tambara-Yamagami categories and Gauss sums

We prove that the higher Frobenius-Schur indicators, introduced by Ng and Schauenburg, give a strong enough invariant to distinguish between any two Tambara-Yamagami fusion categories. Our proofs are based on computation of the higher indicators as quadratic Gauss sums for certain quadratic forms on finite abelian groups and relies on the classification of quadratic forms on finite abelian groups, due to Wall. As a corollary to our work, we show that the state-sum invariants of a Tambara-Yamagami category determine the category as long as we restrict to Tambara-Yamagami categories coming from groups G whose order is not a power of 2. Turaev and Vainerman proved this result under the assumption that G has odd order and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that G does not have a power of 2, cannot be completely relaxed.