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.

Free Lie Algebrassays that $\operatorname{Lie}_n \cong \operatorname{Ind}^{S_n}_{C_n} E$, where $C_n$ is the cyclic group generated by an $n$-cycle in $S_n$, and where $E$ is any faithful $1$-dimensional representation of $C_n$ (that is, the representation which sends said cycle to a primitive $n$-th root of unity). Combining this with the projection formula that is item 3 in mathoverflow.net/q/18799 , we obtain $\operatorname{Lie}_n \otimes \mathbb{C}^n \to \mathbb{C}\left[S_n\right]$ (though, annoyingly, this gives us complex coefficients). $\endgroup$ – darij grinberg Jan 3 '16 at 1:19