Integral almost square-free modular categories

TL;DR

This paper classifies integral almost square-free modular categories with specific Frobenius-Perron dimensions, showing under certain conditions they are group-theoretical, and applies this to categories with odd dimensions less than 1125.

Contribution

It extends classification results for integral almost square-free modular categories and provides conditions under which they are group-theoretical.

Findings

Categories with n ≤ 5 are group-theoretical.

Categories with m prime and m < p are group-theoretical.

Odd-dimensional categories with dimension < 1125 are group-theoretical.

Abstract

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if $n\leq 5$ or $m$ is prime with $m<p$ then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than $1125$ is group-theoretical.