Computations in Symmetric Fusion Categories in Characteristic

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 2nd author) in terms of the usual and super Frobenius-Perron dimensions 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 2 and triangular semisimple Hopf algebras in any characteristic.