Symmetric tensor categories in characteristic 2

Abstract

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k) = p > 0, we use this method to construct generalizations Ver p n , Ver + p n of the incompressible abelian symmetric tensor categories defined in [5] for p = 2 and in [25,27] for n = 1. Namely, Ver p n is the abelian envelope of the quotient of the category of tilting modules for SL 2 (k) by the n-th Steinberg module, and Ver + p n is its subcategory generated by P GL 2 (k)-modules. We show that Ver p n are reductions to characteristic p of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings Z[2 cos(2π/p n )], and that Ver p n embeds into Ver p n+1 . We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Ver p ∞ of the nested sequence Ver p ⊂ Ver p 2 ⊂ · · · . This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of [32], which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in Ver p .

Monoidal abelian envelopes

“…The condition p > 2 is not required and only reflects the limitations of the proof of Lemma 4.3.5 below. Indeed, the equivalent of Theorem 4.3.2 for p = 2 is already known by [BE19]. We start the proof with the following lemma.…”

“…On the other hand, constructing tensor categories with certain requested properties is typically more challenging. In many recent constructions of important new tensor categories (see [BE19, CEH19, CO14, Del07, EHS20]) the desired tensor categories happen to be ‘abelian envelopes’ of straightforward pseudo-tensor categories. We review these examples below via applications of our main result.…”

“…A powerful ‘recognition theorem’ for abelian envelopes was obtained in [EHS20]. However, the construction of abelian envelopes in [BE19, CO14, EHS20] drew from a rich variety of different methods, rather than some standard approach, and moreover at present there is no ‘existence theorem’ in the literature for abelian envelopes.…”

“…The structure theory of tensor categories over fields of positive characteristic is in full development; see for instance [BE19, BEO20, Cou20, EG19, EO19, Ost20]. An important tool developed in [Cou20, EO19, Ost20] is the ‘Frobenius twist’ in arbitrary tensor categories.…”

“…An important tool developed in [Cou20, EO19, Ost20] is the ‘Frobenius twist’ in arbitrary tensor categories. In [BE19] a family of tensor categories in characteristic 2 was constructed in which this functor is not exact. One way to interpret those categories is as the abelian envelopes of the monoidal quotients of the pseudo-tensor category of tilting modules of the reductive group .…”