Lie group SU(n) and symmetric group S$_k$

Question

Given a simple compact Lie group, SU($n$) group, for $n=2,3,4,...$,

Can we determine what are the allowed symmetric groups S$_k$ realized as a subgroup embedded into SU($n$)?

What is the restriction of $k$ for given SU($n$) of

$n=2?$

$n=3?$

$n=4?$

(BONUS but not necessary: Are there some general rules one can use to determine any $n$ what is the allowed $k$?)

Answer 1

In the Euclidean space $\mathbb{R}^n$ you can always construct a standard $n$-simplex $\mathcal{S}_n$ :

https://en.wikipedia.org/wiki/Simplex

This is the convex hull of $n+1$ points. Furthermore one can show that its isometry group is acting $n+1$-transitively and faithfully (it is not so hard to show that you can exchange any two given points and fix every other). As a result its isometry group is $S_{n+1}$ which is therefore included in $SO(n)$ and thus in $SU(n)$ as well.

So, at least, you can always embed $S_{n+1}$ in $SU(n)$. Actually you have the more general rule:

There exists a subgroup of $SU(n)$ isomorphic to $S_{k}$ if and only if $n\geq k-1$.

Remark: $\Leftarrow$ is direct.

Hint: $\Rightarrow$ use character tables for instance.