Symmetric tensor categories in characteristic 2

TL;DR

This paper constructs a sequence of finite symmetric tensor categories over characteristic 2 fields, revealing their incompressibility and algebraic structure related to cyclotomic integers, extending previous specific cases.

Contribution

It introduces a new nested sequence of symmetric tensor categories in characteristic 2, generalizing known examples and analyzing their Grothendieck rings.

Findings

Categories are incompressible and do not admit tensor functors to smaller categories.

Grothendieck rings are isomorphic to rings of real cyclotomic integers.

The sequence generalizes previously known categories by Venkatesh and Ostrik.

Abstract

We construct and study a nested sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category $\mathcal{C}_1$ described by Venkatesh and the category $\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories $\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, $\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.