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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?

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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).

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