TITLE: Young diagram for exterior powers of standard representation of $S_{n}$
QUESTION [13 upvotes]: I'm trying to solve ex. 4.6  in Fulton and Harris' book "Representation Theory". It asks about the Young diagram associated to the standard representation of $S_{n}$ and of its exterior powers. The one of the standard representation $V$ is the partition $\left( n-1, 1 \right)$, while for $\Lambda^{s}V$ is $\left( n-s , 1 , \ldots , 1 \right)$.
For the standard representation I found some hints, such as here, but for the second part I have no clue. The book advises to use Frobenius formula or the branching rule (known also as Pieri's formula)... I tried to think about something like induction on both $n$ and $s$, since the base of the induction for $s$ is given by the standard representation, but then I just stared at the sheet...
Thanks in advance for any hint!
Edit
Maybe I found an explanation, but I'm not sure it is ok!

REPLY [3 votes]: Here is a solution using Pieri's rule:
The representation $\wedge^s V$ has as basis vectors:
$$
\{e_{i_1}\wedge \dotsb \wedge e_{i_s}\mid 1\leq i_1<\dotsb <i_s\leq n\}.
$$
If we restrict this representation to $S_{n-1}$, then the representation on the subspace spanned by $\{e_{i_1}\wedge \dotsb \wedge e_{i_s}\mid 1\leq i_1<\dotsb <i_s\leq n-1\}$ is just the representation of $S_{n-1}$ on $\wedge^s V_{n-1}$, where $V_{n-1}$ is the subspcace of $V$ consisting of vectors with the last coordinate equal to $0$.
On the other hand, the representation of $S_{n-1}$ on the subspace spanned by $\{e_{i_1}\wedge \dotsb \wedge e_{i_{s-1}}\wedge e_n\mid 1\leq i_1<\dotsb <i_{s-1}\leq n-1\}$ is isomorphic to the representation $\wedge^{s-1}V_{n-1}$ of $S_n$.
Therefore, by induction hypothesis, the restriction of $\wedge^s V$ to $S_{n-1}$ is the sum of the representation corresponding to $(n-s, 1^{s-1})$ and the representation corresponding to $(n-s-1, 1^s)$. It follows from Pieri's rule that $\wedge^s V$ is the representation corresponding to $(n-s, 1^s)$.