Title: Braidings on Non-Split Tambara-Yamagami Categories over the Reals

Abstract: In 1998, Tambara and Yamagami investigated fusion categories with a single non-invertible simple object and a straightforward set of fusion rules resulting from self-duality. They classified all possible associators on these categories, thereby classifying all monoidal structures. Two years later, Siehler classified all braiding structures on the same set of fusion rules. This project investigates braidings on a generalization of these fusion rules to a setting where simple objects are no longer required to be split. In particular, we classified braidings on fusion categories over the reals using techniques from the recent paper by Plavnik, Sanford and Sconce that classifies associators on these non-split categories, considering the three possible cases where objects are real, complex or quaternionic. In this talk, we will introduce some key techniques used in our project that allow us to perform graphical computations with string diagrams, and we will demonstrate some examples of these computations before discussing the results.