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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?

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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".

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