# Mathematics:Sandbox

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Let $K$ be a commutative ring. We recall that the $n$-th tensor power $E^{\otimes n}$ of a module $E$ represents the functor sending a module $M$ to the set (in fact $K$-module) of multilinear maps $E^n\to M$. Similarly we can define the $n$-th exterior power $\bigwedge^n(E)$ as the module representing the functor of alternating multilinear maps, and the $n$-th symmetric power $\mathrm{Sym}^n(E)$ as representing the functor of symmetric multilinear maps. The Schur functors generalise these concepts: for each partition $\lambda$ one obtains a module $S^\lambda(E)$ representing a certain class of multilinear maps such that $S^{(1^n)}(E)\cong\bigwedge^n(E)$ and $S^{(n)}(E)\cong\mathrm{Sym}^n(E)$. We can then regard the map $E\mapsto S^\lambda(E)$ as a functor on the category of $K$-modules, even on the category of finitely-generated free $K$-modules.