Schur–Weyl duality over finite fields

Abstract

We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the 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 $${{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}}$$ is an isomorphism. This isomorphism may fail if dim k V is not strictly larger than r.