Representation of $S_n$ that restricts to regular of $S_{n-2}$

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.

• You might be looking for the Whitehouse module (or the Kontsevich module, or the $n$-th component of the cyclic Lie operad). – darij grinberg Nov 26 '17 at 18:11