TITLE: Is there a notation for the symmetric / antisymmetric subspaces of a tensor power that distinguishes them from the symmetric / exterior power?
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: The subspace of $\Sigma_n$-invariants of $V^{\otimes n}$ is called the $n$th divided power of $V$ (at least when $V$ is a free module).