The Schur functor on tensor powers

Abstract

Let M be a left module for the Schur algebra S(n, r), and let s ε ℤ +. Then M ⊗s is a(S(n, rs), FS s)-bimodule, where the symmetric group S sson s letters acts on the right by place permutations. We show that the Schur functor f rs sends M ⊗s to the (FS rs, FS s)-bimodule FS rs⊗ F(SrSs). As a corollary, we obtain the image under the Schur functor of the Lie power L s(M), exterior power s(M)of M and symmetric power S s(M). © 2011 Springer Basel AG.