On tensor autoequivalences of graded fusion categories

Fusion categories generalize the representation theory of finite groups. The simplest examples of fusion categories come from finite groups, their representations, and their cohomology. In general, it is useful to examine group theoretical features of fusion categories such as groups of (isomorphism classes of) tensor invertible objects, and gradings by finite groups. Indeed, every fusion category has a maximal pointed subcategory (generated by tensor invertible objects) and a universal grading by a finite group. We use such features to study tensor autoequivalences. Pointed fusion categories: categories for which all simple objects are tensor invertible, provide our prototype for graded fusion categories. Such categories are well understood; each pointed fusion category has tensor multiplication determined (up to equivalence) by a finite group, and tensor associativity determined (up to equivalence) by a third cohomology class. There is a well known description of the groups of tensor autoequivalences of pointed fusion categories in terms of this data. We use this description to write exact sequences for computing Brauer-Picard groups of pointed fusion categories. We also generalize this description to write exact sequences which describe groups of tensor autoequivalences of graded fusion categories