Semisimplicity in symmetric rigid tensor categories

Let λ be 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 SλV 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∈F(λ) there exist a symmetric rigid tensor category C over k and a non-semisimple object V∈C of dimension d such that SλV 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 SλV is n⋀V or SymnV, and proves a conjecture of Serre (1997) [S2].