Computing Modular Data for Drinfeld Centers of Pointed Fusion Categories Angus Gruen

Angus Gruen's 2017 honours thesis in mathematics, on "Computing Modular Data for Drinfeld Centers of Pointed Fusion Categories", supervised by Scott Morrison, is available here.

### Outline

A theoretical background is developed to explain in detail the link between the modular tensor category $\mathcal Z(\mathsf{Vec}^{\omega}G)$ and the representation category of a quasitriangular quasi Hopf algebra $D^{\omega}\ G$. Using this link, a classification of the simple objects in $\mathcal Z(\mathsf{Vec}^{\omega}G)$ and formulas for the modular data of $\mathcal Z(\mathsf{Vec}^{\omega}G)$ are carefully derived. Then, code is written in GAP to produce the modular data of $\mathcal Z(\mathsf{Vec}^{\omega}G)$, given $\omega$ and $G$. This is used to create a database of modular data for the Drinfeld doubles of pointed fusion categories with dimension less than $47$. This database as well as GAP code accompanying it can be found on this web page. For a basic example of how this database might be used, we briefly analyse patterns in the ranks of $\mathcal Z(\mathsf{Vec}^{\omega}G)$ as $|G|$ varies and produce lower bounds for the number of Morita equivalence classes of pointed fusion categories of a given dimension less than $47$. For dimensions below $32$, these lower bounds agree with the lower bounds published by Mignard and Schauenburg. At dimension $32$ we improve upon the published lower bound and for dimensions $33$ through $47$ we present the first set of lower bounds on the number of Morita equivalence classes of pointed fusion categories at each dimension.