Schur-Weyl duality over finite fields

We prove a version of Schur-Weyl duality over finite fields. We prove that for any field k, if k has at least r+1 elements, then Schur-Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r+1, the natural map kSymr to EndGL(V)(V^otimes r) is an isomorphism. This isomorphism may fail if dimk V is not strictly larger than r.