Morita equivalence of pointed fusion categories of small rank

TL;DR

This paper classifies pointed fusion categories of small rank up to Morita equivalence, revealing that modular data fully distinguish their Morita classes for groups of order less than 32.

Contribution

It provides a complete classification of pointed fusion categories of small rank up to Morita equivalence, using computational tools and modular data as invariants.

Findings

Morita equivalence classes are distinguished by modular data.

Frobenius-Schur indicators alone do not distinguish classes.

Computational methods are developed for cohomology and indicator calculations.

Abstract

We classify pointed fusion categories C(G, $\omega$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $\omega$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued group cohomology and calculating Frobenius-Schur indicators.