Remarks on symmetric fusion categories of low rank in positive characteristic

TL;DR

This paper establishes lower bounds for the rank of symmetric fusion categories in characteristic p, proves properties of the second Adams operation, and classifies categories of small rank with specific features.

Contribution

It provides new bounds on the rank of symmetric fusion categories in positive characteristic and classifies low-rank cases with particular symmetries.

Findings

Second Adams operation is not the identity in non-trivial categories

Classification of rank 3 symmetric fusion categories

Classification of rank 4 categories with two self-dual simple objects

Abstract

We give lower bounds for the rank of a symmetric fusion category in characteristic $p\geq 5$ in terms of $p$. We prove that the second Adams operation $\psi_2$ is not the identity for any non-trivial symmetric fusion category, and that symmetric fusion categories satisfying $\psi_2^a=\psi_2^{a-1}$ for some positive integer $a$ are super Tannakian. As an application, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self dual simple objects.