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 $F{\mathfrak{S}}_{rs}{\otimes }_{F({\mathfrak{S}}_r\wr {\mathfrak{S}}_s)}((f_{r}M{)}^{\otimes s}\otimes F{\mathfrak{S}}_s)$. 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 $M$.

6 pages