Towards higher Frobenius functors for symmetric tensor categories

TL;DR

This paper introduces generalized Frobenius functors for symmetric tensor categories in positive characteristic, expanding their applications in classifying tensor categories and constructing new Verlinde categories.

Contribution

It develops a new theory of monoidal functors generalizing Frobenius functors and constructs generalized Verlinde categories using elementary abelian p-groups.

Findings

New construction of Verlinde categories $Ver_{p^n}$

Families of functors conjectured to relate to $Ver_{p^n}$

Partial evidence that these functors detect categories fibered over $Ver_{p^n}$

Abstract

We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from \cite{Os, EOf, Tann}. The latter has proved to be a powerful tool in the ongoing classification of tensor categories of moderate growth, and we demonstrate the similar potential of the generalisations. More explicitly, we describe a new construction of the generalised Verlinde categories $Ver_{p^n}$ in terms of representation categories of elementary abelian $p$-groups. This leads to families of functors relating to $Ver_{p^n}$ that we conjecture, and partially show, to exhibit the characteristic properties of the Frobenius functor relating to $Ver_p$. In particular, we conjecture some of these functors to detect categories that fibre over $Ver_{p^n}$.