Abstract: In characteristic zero, every symmetric tensor category of moderate growth fibres over super vector spaces. This means that any such category is equivalent to a category of representations of some super group, as a consequence of Tannakian formalism. Moreover, the parity of an object in such a category can be determined by its symmetric and exterior powers, with even and odd objects having non-vanishing symmetric and exterior powers respectively. However, in positive characteristic there are tensor categories which do not have such a fibre functor, and classifying these categories is still an open problem. Objects in these categories can have many unusual properties, including having finite symmetric and exterior algebras simultaneously.

 

In this talk, I will discuss recent work with Kevin Coulembier and Pavel Etingof in which we classify all symmetric tensor categories generated by an object with minimal symmetric and exterior algebras. Along the way I will cover several important concepts in the study of symmetric tensor categories, including Tannaka-Krein duality and Schur-Weyl duality, and how they apply to Verlinde categories of algebraic groups.