I am looking for an example or some intuition as to why for the standard representation $V$ of the unitary group $\wedge^nV$ is irreducible while, according to exercise 6.16 in Fulton and Harris’s book on representation theory, the isomorphic vector space $(\wedge^{n/2}V)\wedge (\wedge^{n/2}V)$, for simplicity even $n$, is not irreducible. It appears, by Schur-Weyl duality, the latter space is the direct sum of the images of Schur functors corresponding to irreps of the symmetric group with partitions that contain $n-a$ $2$‘s and $2a$ $1$’s for odd $a$. I believe these spaces correspond to strictly lower than $n$ alternating powers of $V$.
I’m guessing that this has to do with no equivariant maps existing between the two spaces? Although, I do not see that such a map does not exist. Specifically, it seems that the action of the unitary group induced by the tensor power of the standard representation is the same for both spaces. My question is also for the symmetric product $\text{Sym}^{n}V$ as well.