Vol. 9, No. 8, 2015

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Indicators of Tambara–Yamagami categories and Gauss sums

Tathagata Basak and Ryan Johnson

Vol. 9 (2015), No. 8, 1793–1823
DOI: 10.2140/ant.2015.9.1793

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 in terms of Gauss sums for certain quadratic forms on finite abelian groups and rely 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| is not a power of 2 cannot be completely relaxed.

fusion category, Tambara–Yamagami category, Frobenius–Schur indicator, discriminant form, quadratic form, Gauss sum
Mathematical Subject Classification 2010
Primary: 18D10
Secondary: 15A63, 11L05, 57M27
Received: 24 July 2014
Revised: 18 June 2015
Accepted: 25 July 2015
Published: 29 October 2015
Tathagata Basak
Department of Mathematics
Iowa State University
Ames, IA 50011
United States
Ryan Johnson
Department of Mathematics
Grace College
Winona Lake, IN 46590
United States