On Frobenius exact symmetric tensor categories

Abstract

A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic $p$ version of this theorem. Namely, we show that a pre-Tannakian category over an algebraically closed field of characteristic $p>0$ admits a fiber functor into the Verlinde category $\mathrm{Ver}_p$ (i.e., is the representation category of an affine group scheme in $\mathrm{Ver}_p$) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderategrowth has a fiber functor to $\mathrm{Ver}_p$ (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics $2$,$3$). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation $V$, the possible growth rates of the number of indecomposable summands in $V^{\otimes n}$of dimension prime to $p$.