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 $F\mathfrak{S}_{rs} \otimes_{F(\mathfrak{S}_r \wr \mathfrak{S}_s)} ((f_rM)^{\otimes s} \otimes F\mathfrak{S}_s)$. As a corollary, we obtain the effect of the Schur functor on the Lie power $L^s(M)$, symmetric power $S^s(M)$ and exterior power $\bigwedge^s(M)$ of $M$.