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.
