N-dimensional linear representation of symmetric group S(N)

Question

A beginner's question on the linear representations of the symmetric group $S(n)$: playing around with GAP, e.g.:

gap> CharacterDegrees(SymmetricGroup(n));

for various values of $n$, it appears that, no irreducible representation of dimension $n$ exists, except if $n = 5$. In fact, there are two 5-dimensional representations of $S(5)$. My questions:

1. Is it correct that $S(n)$ has no irreducible representation of dimension $n$, except if $n = 5$? If yes, what is special about $S(5)$ that is allows a 5-dimensional representation?
2. How do I construct the 5-dimensional representations of $S(5)$?

Thanks

Answer 1

The group $S_5$ has 6 Sylow 5-subgroups, so it acts on them by conjugation. This gives an embedding $S_5 \hookrightarrow S_6$ which turns out not to be conjugate to the "standard embedding". There are two irreducible representations of $S_6$ given by the standard representation and its twist by the sign, and these turn out to restrict to irreducible representations of $S_5$.

You are right that no other symmetric group has this property -- if I understand the Wikipedia article right, this follows from the full classification of representations of the symmetric group in terms of Young diagrams, but there might be an easier proof I don't know.`