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Abstract: Let \lambda be a partition of a positive integer n. Let C be a symmetricrigid tensor category over a field k of characteristic 0 or chark>n, and letV be an object of C. In our main result Theorem 4.3 we introduce a finite setof integers F\lambda and prove that if the Schur functor \mathbb{S} {\lambda}V of V is semisimple and the dimension of V is not in F\lambda, then V issemisimple. Moreover, we prove that for each d in F\lambda there exist asymmetric rigid tensor category C over k and a non-semisimple object V in C ofdimension d such that \mathbb{S} {\lambda} V is semisimple which shows thatour result is the best possible. In particular, Theorem 4.3 extends twotheorems of Serre for C=RepG, G is a group, and \mathbb{S} {\lambda} V is\wedge^n V or Sym^n V, and proves a conjecture of Serre \cite{s1}.



Author: Shlomo Gelaki

Source: https://arxiv.org/



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