On the Frobenius functor for symmetric tensor categories in positive characteristic

Pavel Etingof 1  and Victor Ostrik 2
  • 1 Department of Mathematics, Massachusetts Institute of Technology, MA 02139, Cambridge, USA
  • 2 Department of Mathematics, University of Oregon, OR 97403, Eugene, USA
Pavel Etingof
  • Corresponding author
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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and Victor Ostrik
  • Department of Mathematics, University of Oregon, Eugene, OR 97403, USA, Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia
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Abstract

We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 over a field 𝒌 of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F:𝒞𝒞Verp, where Verp is the Verlinde category (the semisimplification of Rep𝐤(/p)); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 is finite and semisimple, then it admits a fiber functor to Verp. The main new feature is that when 𝒞 is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞Verp. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞ex inside any STC 𝒞 with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Verp. This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞ex. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌[d]/d2 with d primitive and R-matrix R=11+dd), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

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