Reflection fusion categories

TL;DR

This paper introduces reflection fusion categories generated by specific objects and establishes a correspondence with orthogonal reflection groups over finite fields, enabling their classification using known group classifications.

Contribution

It defines reflection fusion categories and links their classification to that of orthogonal reflection groups over finite fields, providing a new framework for understanding these categories.

Findings

Reflection fusion categories correspond to orthogonal reflection groups over _p.

Classification of irreducible reflection fusion categories is achieved via known reflection group classifications.

Categories are generated by objects of Frobenius-Perron dimensions 1 and _p.

Abstract

We introduce the notion of a $\textit{reflection fusion category}$, which is a type of a $G$-crossed category generated by objects of Frobenius-Perron dimension $1$ and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories correspond to orthogonal reflection groups over $\mathbb{F}_p$. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories.