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**Drexel28** So, we know that if $\displaystyle \mathcal{V}$ is a finite dimensional [mat]F[/tex]-space and $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ is the set of $\displaystyle n$-forms on $\displaystyle \mathcal{V}$ that $\displaystyle S_n$ acts on $\displaystyle \text{Mult}_n\left(\mathcal{V}\right)$ via the action $\displaystyle \left(\pi K\right)(x_1,\cdots,x_n)=K\left(x_{\pi(1)},\cdots, x_{\pi(n)}\right)$.

So, this might be a stupid question...but besides being able to state things easier (the symmetric forms are exactly the kernel of the action) how does this fact really help us? All the theorems I know which are useful and pertain to group actions apply for finite sets only.