Integral modular categories of Frobenius-Perron dimension $pq^n$

TL;DR

This paper classifies integral modular categories with Frobenius-Perron dimension of the form pq^n, showing they are mostly group-theoretical or contain specific Tannakian subcategories, with new results for dimensions up to pq^7.

Contribution

It extends classification results for integral modular categories of dimension pq^n, identifying conditions under which they are group-theoretical or contain Tannakian subcategories.

Findings

Categories with dimension pq^5 are group-theoretical.

Categories with dimension pq^6 or pq^7 (p<q) are group-theoretical.

Generalized criterion for integral modular categories to be group-theoretical.

Abstract

Integral modular categories of Frobenius-Perron dimension $pq^n$, where $p$ and $q$ are primes, are considered. It is already known that such categories are group-theoretical in the cases of $0 \leq n \leq 4$. In the general case we determine that these categories are either group theoretical or contain a Tannakian subcategory of dimension $q^i$ for $i>1$. We then show that all integral modular categories $\mathcal{C}$ with $\mathrm{FPdim}(\mathcal{C})=pq^5$ are group-theoretical, and, if in addition $p<q$, all with $\mathrm{FPdim}(\mathcal{C})=pq^6$ or $pq^7$ are group-theoretical. In the process we generalize an existing criterion for an integral modular category to be group-theoretical.