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

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.