Computing the center of a fusion category

TL;DR

This paper introduces an algorithm for computing the Drinfeld center of pivotal fusion categories, implemented in software, enabling explicit calculations of objects, half-braidings, and related symbols for categories up to rank 6.

Contribution

The authors develop a novel algorithm and software implementation for explicitly computing the Drinfeld center of pivotal fusion categories, including their $F$- and $R$-symbols.

Findings

Successfully computed centers for 279 categories up to rank 5.

Generated explicit models of centers with objects and half-braidings.

Extended computations to select rank 6 categories, including the Haagerup subfactor.

Abstract

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,\gamma)$ where $X$ is an object and $\gamma$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper).