Let V be a finite dimensional vector space and consider $(Sym^n V^\vee)^\vee$ where $\vee$ denotes thedual, i.e homogenous polynomials in V of degree n. Consider as well $S_n(V)$, consisting of fixed points of the canonical action of $\Sigma_n$ , the nth symmetric group on $V^{\otimes n}$. I am trying toshow that $S_n(V)$ and $(Sym^n V^\vee)^\vee$are isomorphic but no luck. Could someone help me?
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Hint: Use $(Sym^nV^{\vee}) \times S_n(V) \to k$ where $k$ is the ground field and is defined as $(f,v) \mapsto f(v).$ – Ehsan M. Kermani Apr 20 '14 at 18:19
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@EhsanM.Kermani What is the pairing and why is it welldef. and an Iso? – Marvelaction Apr 20 '14 at 23:26