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.



factorization of the regular representation of the symmetric group
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.)


