The Schur functor on tensor powers

Abstract

Let M be a left module for the Schur algebra S(n, r), and let ${s \in \mathbb{Z}^+}$ . Then ${M^{\otimes s}}$ is a ${(S(n,\,rs), F{\mathfrak{S}_{s}})}$ -bimodule, where the symmetric group ${{\mathfrak{S}_s}}$ on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends ${M^{\otimes s}}$ to the ${(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}$ -bimodule ${F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}$ . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s(M), exterior power ${\bigwedge^s(M)}$ of M and symmetric power S ^s(M).