Introduced by Pavel Etingof, Dmitri Nikshych, and Victor Ostrik (2005), "group-theoretical fusion categories" (GTFCs) are a certain kind of monoidal category whose construction depends on group-theoretic data. They are a vital part of the classification program of general fusion categories, and due to their explicit construction, they also serve as a go-to testing ground for results about fusion categories. The goal of this talk is to present results on the representation theory of GTFCs, which by work of Ostrik (2003), boils down to understanding algebraic structures in GTFCs. The main result is a construction of explicit Morita equivalence class representatives of indecomposable, separable algebras in GTFCs. Background on monoidal categories will be provided, along with interesting auxiliary results appearing in our work. This is joint work with Yiby Morales, Monique Müller, Julia Plavnik, Ana Ros Camacho, and Angela Tabiri.
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Algebraic Structures In Group-Theoretic Fusion Categories