New incompressible symmetric tensor categories in positive characteristic

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, then we use this method to construct the incompressible abelian symmetric tensor categories Verpn, Verpn+ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for n=1, and by Benson–Etingof for p=2. Namely, Verpn is the abelian envelope of the quotient of the category of tilting modules for SL2(k) by the nth Steinberg module, and Verpn+ is its subcategory generated by PGL2(k)-modules. We show that Verpn are reductions to characteristic p of Verlinde braided tensor categories in characteristic 0, 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[2cos(2π∕pn)], and that Verpn embeds into Verpn+1. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Verp∞ of the nested sequence Verp⊂Verp2⊂⋯ . This would provide an analogue of Deligne’s theorem in characteristic 0 and a generalization of the results of Coulembier, Etingof, and Ostrik, which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and, moreover, the fiber functor lands in Verp (in the case of fusion categories, this was shown earlier by Ostrik). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories Verpn.