Computations in symmetric fusion categories in characteristic p

Abstract

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde fiber functor $F(X)$ of $X$ (defined by the second author) in terms of the usual and super Frobenius-Perron dimension of $X$. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic.