On the Equivalence of Module Categories over a Group-Theoretical Fusion Category

TL;DR

This paper provides a cohomological criterion for classifying indecomposable module categories over group-theoretical fusion categories, completing their classification.

Contribution

It establishes a necessary and sufficient condition based on group cohomology for the equivalence of module categories over such fusion categories.

Findings

Cohomological criterion for module category equivalence

Complete classification of indecomposable module categories

Clarification of the structure of group-theoretical fusion categories

Abstract

We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module categories.