On the Brauer–Picard groups of fusion categories
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We develop methods of computation of the Brauer–Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac–Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer–Picard groups of pointed fusion categories.
We are grateful to Derek Holt, Geoffrey Mason, Victor Ostrik, and Leonid Vainerman for helpful discussions. We thank Cesar Galindo for useful discussions and for pointing the paper  to us. We thank Ehud Meir for explaining to us the classification of module categories over Tambara–Yamagami categories from  and sharing his calculations with us. Finally, we thank the anonymous referee for carefully reading the paper and suggesting many improvements of the exposition. The authors were partially supported by the NSA Grant H98230-15-1-0003. The second author was partially supported by the NSA Grant H98230-16-1-0008.
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