Misc,
On Frobenius exact symmetric tensor categories
(2021)cite arxiv:2107.02372Comment: With an appendix by Alexander Kleshchev.
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 Ver_p (i.e., is the representation category of an affine group scheme
in 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 moderate growth has
a fiber functor to 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^n of dimension prime to p.
