Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines two subspaces of $V^{\otimes n}$, the subspace on which $S_n$ acts via the trivial character and the subspace on which $S_n$ acts via the antisymmetric character.
Question 0: Is the construction of these subspaces functorial in $V$? If it is, are the corresponding functors naturally isomorphic to the symmetric and exterior powers, and if that's true, are the corresponding natural isomorphisms unique?
If the answers to Question 0 turn out more or less like I suspect, we should not regard these subspaces as completely synonymous with the symmetric power $S^n V$ and the exterior power $\Lambda^n V$, respectively, since these are naturally thought of as quotients of $V^{\otimes n}$. (This issue recently came up in another MO question.)
Question 1: Is there an established notation in the literature which respects this distinction?
Best Answer
This only answers part of your question 0 unfortunatly. The construction is certainly functorial, but the two notions of symmetric/alternating power do not always agree. Let's write $\operatorname{Sym}^n (V)$ for the symmetric tensors, and $\operatorname{Alt} ^n (V)$ for the alternating tensors. I wish this were established notation, but it probably isn't. Let $p$ be the characteristic of the field. Note $V^{\otimes n}$ is a $kGL(V) - \Sigma_n$ bimodule ($\Sigma_n$ is the symmetric group). Then if $r$ is less than $p$, or if $p=0$, $\operatorname{Sym}^r (V) \cong S^r(V)$ and $\operatorname{Alt}^r (V) \cong \Lambda ^r(V)$ as $GL(V)$-modules (this is proved by writing down maps explicitly).
If $r \geq p$ then $S^r$ and $\operatorname{Sym}^r$ are the contravariant (i.e. transpose) duals of one another as $GL$ modules. I imagine the same is true of the alternating power/antisymmetric tensors.