TITLE: factorization of the regular representation of the symmetric group
QUESTION [15 upvotes]: Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question:  Does there exist a representation $V$ (of dimension $(n-1)!$) such that $V\otimes\mathbb{C}^n\cong \mathbb{C}[S_n]$?  If so, does $V$ admit any particularly nice description?
For example, when $n=4$, we can take $V = V(4)\oplus V(2,1,1) \oplus V(2,2)$.
This problem has a nice solution if we only ask the isomorphism to be equivariant for the subgroup $S_{n-1}\subset S_n$.  We can realize $\mathbb{C}[S_n]$ as the cohomology of the configuration space of $n$ points in $\mathbb{R}^k$ for any odd $k$ (even $k=1$ is okay).  Then forgetting the $n$th point gives us a fiber bundle that behaves like a product on cohomology, and we obtain an isomorphism $\mathbb{C}[S_n]\cong \mathbb{C}[S_{n-1}]\otimes\mathbb{C}^n$.  But we have broken the symmetry by choosing which point to forget, so this is only an isomorphism of $S_{n-1}$ representations.  I want to do everything $S_n$-equivariantly, which is indeed possible when $n=4$, as the above example demonstrates.
Update:  As Geoff Robinson observes in the comments below, an $S_n$-representation $V$ is a solution to my problem if and only if the restriction of $V$ to $S_{n-1}$ is isomorphic to the regular representation.

REPLY [8 votes]: Let $H$ be a regular subgroup of $S_n$, for instance a transitive cyclic subgroup of order $n$. Then the permutation module $V$ of the action on the coset space $S_n/H$ has the requested property, as only the identity element of $S_n$ lies in a conjugate of $H$ and fixes a point in the natural action at the same time.
(Remark: Just noticed that in the very same minute Geoff Robinson gave the identical answer in a comment. So I marked my answer CW.)

REPLY [7 votes]: The fact that the Lie module (as proposed by Darij Grinberg) works, as well, as an explicit isomorphism of modules, follows from the theory of cyclic operads: see Corollary 6.9 in http://sites.math.northwestern.edu/~getzler/Papers/cyclic.pdf (to be precise, restrict the statement of that Corollary to $S_n\subset S_{n+1}$).