On Frobenius exact symmetric tensor categories (2107.02372v2)

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^{\otimes n} of dimension prime to p.

Frobenius Exact Symmetric Tensor Categories in Positive Characteristic

The paper "On Frobenius Exact Symmetric Tensor Categories", authored by Kevin Coulembier, Pavel Etingof, and Victor Ostrik, ventures into expanding the theoretical framework of tensor categories. The authors work to extend Deligne’s theorem about pre-Tannakian categories in characteristic zero to those in characteristic $p > 0$. Centered on symmetric tensor categories over an algebraically closed field of positive characteristic, the paper seeks to provide a characterization of pre-Tannakian categories, focusing on conditions such as moderate growth and Frobenius exactness.

The principal result of the paper establishes that pre-Tannakian categories over an algebraically closed field of characteristic $p > 0$ admit a fiber functor into the Verlinde category if and only if they are Frobenius exact and of moderate growth. This work answers a conjecture posed by one of the authors in 2015, offering insights into the dimension theory within symmetric tensor categories under positive characteristic conditions.

One significant contribution of the paper is showing that Frobenius exact pre-Tannakian categories of moderate growth admit a well-defined notion of Frobenius-Perron dimension. This generalizes the concept of Frobenius-Perron dimension traditionally applicable only to finite tensor categories, opening avenues in representing categories of affine group schemes. By extending dimension theory in this setting, the paper paves the way for new methodologies in examining modular representation theories, which can traditionally be intractable with conventional techniques.

Central to the proofs are the notions of Frobenius functors and their properties, particularly in the novel setting of positive characteristic. The authors introduce the concept of 'enhanced Frobenius functors,' discussing their exactness and the criteria under which pre-Tannakian categories maintain Frobenius exactness. The paradigm is augmented through a stabilization procedure, showing how symmetric tensor categories can embed faithfully into categories with bijective Frobenius functors.

Applications of these theoretical advancements extend into the classification of semisimple pre-Tannakian categories in positive characteristic, building on existing principles from modular representation theory and p-adic dimensions. The results apply to understanding growth rates in modular representation theory by characterizing possible growth rates of indecomposable summands.

Moreover, some conjectures related to the properties of symmetric tensor categories in positive characteristic are discussed, which could redefine our understanding of tensor product decompositions and pave the way for future research endeavors. The possible uniform boundedness of Frobenius-Perron dimensions, along with algebraic dimensions remaining consistent even outside Frobenius exact contexts, offers ample prospects for new inquiries.

In summary, this paper intricately extends the field of symmetric tensor categories into positive characteristic, enriching the landscape with new constructs like Frobenius-Perron dimensions in infinite categories and presenting a robust theoretical backing for further explorations in modular representation theory.