# Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$?

Deligne's theorem on tensor categories states that for any symmetric tensor category $$\mathcal{C}$$ satisfying the subexponential growth condition, there is a fiber functor to $$\mathsf{sVec}$$ and that $$\mathcal{C}$$ is equivalent to the representations of a supergroup. Under what conditions can we draw the stronger conclusion that $$\mathcal{C}$$ is equivalent to $$\operatorname{\mathsf{Rep}}G$$ for some group $$G$$? Does it suffice for the simple objects in $$\mathcal{C}$$ to all have positive dimension?

## 1 Answer

If you are in characteristic zero and the dimension of every object is a positive integer then it admits a fiber functor to $$Vec$$ and is equivalent to $$Rep(G)$$ for some (pro-algebraic) group $$G$$. This is theorem 7.1 in Deligne's "Categories Tannakiennes".