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Semisimplicity in symmetric rigid tensor categories

Let λbe a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of integers F(λ) and prove that if the Schur functor \mathbb{S}_λ V of V is semisimple and the dimension of V is not in F(λ), then V is semisimple. Moreover, we prove that for each d in F(λ) there exist a symmetric rigid tensor category C over k and a non-semisimple object V in C of dimension d such that \mathbb{S}_λ V is semisimple (which shows that our result is the best possible). In particular, Theorem 4.3 extends two theorems of Serre for C=Rep(G), G is a group, and \mathbb{S}_λ V is \wedge^n V or Sym^n V, and proves a conjecture of Serre (\cite{s1}).