In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special case to the following (conjectural, to me) isomorphism
$$
\operatorname{Sym}^k V \cong \left[\bigwedge^k (\mathbb C^k \otimes V)\right]^{SL(k)} 
$$
Here $k$ is a positive integer, $\mathbb C^k$ is the standard representation of $SL(k)$, and on the right I am taking invariants of the induced action, $V$ is any $\mathbb C$ vector space (in our case $V$ has dimension three, but I am guessing that is not important).


I think that the map might be something like
$$
v\_1 \otimes \cdots \otimes v\_k \mapsto \sum\_{\sigma \in S\_k} \operatorname{sgn}(\sigma)(e\_{\sigma(1)} \otimes v\_1)\wedge \cdots \wedge (e\_{\sigma(k)} \otimes v\_k)
$$ 
(Here $e\_j$ is a basis for $\mathbb C^k$, and $S\_k$ is the symmetric group.)


I can at least check that the right hand side is invariant, (carefully for $k=2$, and it seems good for bigger $k$ also).


It seems like this might be a standard result (if it is true), but I don't know enough about the subject to know where to look. Can anyone point me in the right direction?



Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?
Yes, this is correct. Let me hit it with a more general statement in case that becomes useful for further generalizations.


There is a general formula (Cauchy identity) for the action of $GL(V) \times GL(W)$ on the exterior power $\bigwedge^n(V \otimes W)$. This is written as


$\bigoplus\_{|\lambda|=n} S\_\lambda(V) \otimes S\_{\lambda^\dagger}(W)$


where $S\_\lambda$ is a Schur functor associated to a partition of size $n$, and $\lambda^\dagger$ corresponds to its transpose. In your situation when $n = \dim(V)$, the only possible Schur functor that corresponds to invariants is $\lambda = (1,1,\dots,1)$ where $S\_\lambda$ is the exterior power also. In that case, $\lambda^\dagger = (k)$ and the corresponding Schur functor is the symmetric power.


There are many references for this statement -- one possibility is Weyman's book, Cohomology of Vector Bundles and Syzygies (Chapter 2), which treats all of this from scratch (though I admit what I am saying is most likely overkill for your question).


