On Frobenius exact symmetric tensor categories

Abstract

We apply the recently introduced notion, due to Dyckerhoff, Kapranov and Schechtman, of N -spherical functors of stable infinity categories, which generalise spherical functors, to the setting of monoidal categories. We call an object N -bounded if the corresponding regular endofunctor on the derived category is N -spherical. Besides giving new examples of N -spherical functors, the notion of N -bounded objects gives surprising connections with Jones-Wenzl idempotents, Frobenius-Perron dimensions and central conjectures in the field of symmetric tensor categories in positive characteristic.

Growth rates of the number of indecomposable summands in tensor powers

“…With further work, one can even construct an example which satisfies all conditions except the abelian condition 2.2.1(1). Now we reformulate some results from [CEO,De1].…”

“…Proof. Part (2) follows from [CEO,Lemmata 8.3 and 8.5]. In particular, [CEO,Lemma 8.5] states that gd(F X) = gd(X).…”

“…Proof. The naive filtered colimit of a system of tensor categories is canonically again a tensor category, see for instance [CEO,§6]. In [CEO,Lemma 6.3], it is proved that such filtered colimits of categories MdGr k remain in MdGr k .…”

“…Part (1) follows from Lemma 2.2.8. It follows from [CEO,Lemma 4.9] that every target of a surjective tensor functor out of C has again moderate growth. Part (2) thus follows from Proposition 5.1.4(1).…”

“…For k of characteristic p > 0, the subject has developed rapidly in the last few years, see for instance [BE1,BEO,CEO,Co1,EG1,EG2,EO,Os]. An analogue of Deligne's theorem for Frobenius exact tensor categories (meaning the Frobenius functor from [Co1,EO] is exact) was obtained in [CEO], where one has to replace the category of supervector spaces with the larger Verlinde category Ver p .…”