I look for a representation $V$ of dimension $n-2!$ (over complex numbers) of the symmetric group $S_n$, such that the restriction to $S_{n-2}$ is the regular representation of $S_{n-2}$.
There a such representation exists? Is it (almost) unique? How does $V$ decompose as sum of irreducible representations $V_\lambda$ for $\lambda$ partition of $n$?
Suppose that $V$ satisfies the condition above so does the representation $V \otimes sgn$. For $n=4$ the representation exists, is unique and is the irreducible representation $V_{(2,2)}$.
Thanks in advance. For $n=5$ both $$V_{(2,2,1)} \oplus V_{(5)} $$ and $$V_{(3,2)} \oplus V_{(1,1,1,1,1)} $$ restricts to the regular representation of $S_{n-2}$, by the Pieri rule.
We can observe that the standard representation $V_{(n-1,1)}$ and $V_{(n-1,1)}\otimes sgn$ cannot be direct addendum of $V$, because their restriction to $S_{n-2}$ contain two copies of the trivial (respectively, the sign $sgn$) representation.
Thanks in advance!