Geometric and Modular Representation Theory Seminar

I will report on a joint work in progress with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic $p > 0$ admits a fiber functor to the Verlinde category (semisimplification of $Rep(Z/p)$) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for $p = 2$ and $3$ this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for $p > 3$. This generalizes Deligne's theorem that any symmetric tensor category over $C$ of moderate growth is super-Tannakian to characteristic $p$. At the end I'll discuss applications of this result to modular representation theory.