Are representations of symmetric groups determined by its restrictions?

Question

Are there two representations $U$ and $V$ of the symmetric group $S_n$, such that $Res_{S_{n-1}}^{S_n} U\cong Res_{S_{n-1}}^{S_n} V$, but $U$ and $V$ are not isomorphic as representations of $S_n$? I am aware that the branching rule reduces this question to a combinatorial question about Young diagram. But the answer to that combinatorial question is equally unclear to me. Thank you!

Answer 1

Yes this is entirely possible. Irreducible representations of the symmetric group are in bijection with Young diagrams $\lambda \vdash n$ (equivalently, partitions of $n$). Let $V_\lambda$ denote the irreducible representation corresponding to the partition $\lambda$. Consider the following example.

1) $\lambda = [3]$: $V_{[3]}$ is the trivial representation of $S_3$
2) $\lambda = [1,1,1]$: $V_{[1,1,1]}$ is the sign representaion of $S_3$
3) $\lambda = [2,1]$: $V_{[2,1]}$ is the standard representation of $S_3$

The `Pieri rule' tells you how to restrict these representations, and you will see that $V_{[3]} \oplus V_{[1,1,1]}$ and $V_{[2,1]}$ both restrict to $V_{[2]} \oplus V_{[1,1]}$.

Answer 2

Even more basic example: The trivial and sign representations of $S_2$ restrict to the same representation of $S_1$.

(In Young Diagram notation, these are respectively $V_{[2]}$ and $V_{[1, 1]}$ restricting to $V_{[1]}$.)