Finite symmetric algebras in tensor categories and Verlinde categories of algebraic groups

TL;DR

This paper classifies objects in symmetric tensor categories with finite symmetric and exterior powers, revealing their connection to Verlinde categories of algebraic groups in positive characteristic.

Contribution

It provides a complete classification of such objects when the sum of degrees equals the characteristic p, linking them to Verlinde categories of reductive groups.

Findings

Objects with finite symmetric and exterior powers exist only in characteristic p.

All such objects correspond to Verlinde categories of reductive algebraic groups.

The classification fills gaps in the understanding of these tensor categories.

Abstract

We investigate objects in symmetric tensor categories that have simultaneously finite symmetric and finite exterior algebra. This forces the characteristic of the base field to be $p>0$, and the maximal degree of non-vanishing symmetric and exterior powers to add up to a multiple of $p$. We give a complete classification of objects in tensor categories for which this sum equals $p$. All resulting tensor categories are Verlinde categories of reductive groups and we fill in some gaps in the literature on these categories.