Proving that a representation is irreducible

Question

Let $V_n = \mathbb{C}^n$ the representations of $SU(n)$ given by matrix multiplication $SU(n) \times V_n \rightarrow V_n, (A, v) \mapsto A \cdot v .\,$ Show that $V_n$ is irreducible.

I tried to prove this by induction using the fact that the representation is completely reducible, because $SU(n)$ is a compact Lie group, but I'm not sure is the right way to do it.

Any suggestions? Thanks in advance!

Answer 1

Let $v\in V_n\setminus\{0\}$. For each $w\in V_n$ such that $\|w\|=\|v\|$, there is a $M\in SU(n)$ such that $M.v=w$. Therefore, if $U\subset V_n$ is a vector subspace such that

• $v\in U$;
• $SU(n).U\subset U$,

then $U\supset\{w\in V_n\,|\,\|w\|=\|v\|\}$. But then $U=V_n$.