Let $V$ be a finite dimensional vector space. Let $W$ be a representation of $GL\left( V \right)$ with a homomorphism $f:S^2W\rightarrow \bigwedge ^n V$ as $GL\left( V \right)$ representations, what is $W$ and $f$?
If $n$ is a multiple of 4, $n=4k$, it seems that $W=\bigwedge^{2k} V$ seems to work, i.e. $f:S^2\bigwedge^{2k} V\rightarrow \bigwedge^{4k} V$ where $f:xy\mapsto x\land y$. Are there other solutions? Are there solutions with $n\neq 4k$?
If $S^2W$ decomposes into irreducible subreprentations, then it seems we should have $S^2W\cong \bigwedge ^n V \oplus \ker f$, but I'm not sure where to go from here. It seems that finding solutions should have something to do with plethysm and Schur Weyl duality, but I'm not clear about these things right now.