On the Frobenius functor for symmetric tensor categories in positive characteristic

We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ is finite and semisimple then it admits a fiber functor to ${\rm Ver}_p$. The main new feature is that when $\mathcal{C}$ 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 $\mathcal{C}\to {\rm Ver}_p$. 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 $\mathcal{C}_{\rm ex}$ inside any STC $\mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by $F$. We prove that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to ${\rm Ver}_p$. We also show that a sufficiently large power of $F$ lands in $\mathcal{C}_{\rm ex}$. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category and show that a STC with Chevalley property is (almost) Frobenius exact.