Computations in symmetric fusion categories in characteristic p

TL;DR

This paper investigates symmetric fusion categories over fields of characteristic p, introducing super Frobenius-Perron dimensions, deriving explicit formulas for the Verlinde fiber functor, and classifying certain low-rank categories and Hopf algebras.

Contribution

It introduces the super Frobenius-Perron dimension, provides explicit formulas for the Verlinde fiber functor, and classifies rank two braided fusion categories and triangular semisimple Hopf algebras in characteristic p.

Findings

Derived an explicit formula for the Verlinde fiber functor in terms of super Frobenius-Perron dimensions.

Computed the decomposition of symmetric powers in Verlinde categories, generalizing classical invariant formulas.

Proved the uniqueness of the Verlinde fiber functor and classified certain low-rank categories and Hopf algebras.

Abstract

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde fiber functor $F(X)$ of $X$ (defined by the second author) in terms of the usual and super Frobenius-Perron dimension of $X$. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic.