Finite symmetric algebras in tensor categories and Verlinde categories of algebraic groups
Kevin Coulembier, Pavel Etingof, Joseph Newton
TL;DR
The paper classifies objects X in symmetric tensor categories over a field of positive characteristic p that have both finite symmetric and finite exterior algebras, focusing on the extremal case where the largest nonzero degrees satisfy m(X)+n(X)=p. It develops a two-pronged approach: (i) a detailed analysis of symmetric/exterior powers, filtrations, and their behavior under exact sequences, leading to the notion of N(X)=m(X)+n(X); (ii) a Lie-theoretic reduction showing that any such category is governed by Verlinde categories Ver_p(G) for reductive G, via principal SL_2 embeddings and the structure Ver_p(G)≅Ver_p^+(G)⊠Ver_p^0(G). The main result classifies all possibilities when N(X)=p, showing the ambient category must be one of Ver_p(G) for specific adjoint-type groups (e.g., PGL_n, SO_n, G_2, E_7), with precise constraints on the image of X in Ver_p^+(G). This yields a complete picture of semisimple tensor categories generated by a non-invertible object with the extremal finite-powers property and clarifies the equivalences among Verlinde categories arising from different simple groups. The work also establishes a broader decomposition principle for Ver_p(G) and broadens the literature on Verlinde categories by filling gaps in their construction and equivalences.
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.