For an irrep of $G$ (dimension larger than 1) we have that $P=\frac{1}{|G|}\sum_{g\in G}g$ acts as $0$

Question

It was a long time ago that I did representation theory, so now I am struggling with a simple question. We have a finite group $G$ and an irrep $\rho\colon \mathbb{C}[G]\rightarrow \mathrm{End}(V)$ of $\mathbb{C}[G]$ which has dimension at least 2. Consider now the element $P:=\frac{1}{|G|}\sum_{g\in G}g$. It is easily seen that $P^2=P$ and that $P$ is in the center of $\mathbb{C}[G]$. Hence the linear map $\rho(P)\colon V\rightarrow V$ is an intertwining map. By Schurs lemma, either $\rho(P)=0$ or $\rho(P)=\lambda\mathrm{Id}$ (this $\lambda=\pm 1$ which follows from $P$ being idempotent). I want to show that when $\mathrm{dim}(V)>1$ that we are always in the first case, i.e. $\rho(P)=0$. I need some hints, thanks.

Answer 1

Let $v\in V$ be nonzero. If $Pv\neq 0$, then $Pv$ spans a $1$-dimensional sub-representation of $V$ (since $gP=P$ for any $g\in G)$. If $V$ is irreducible of dimension greater than $1$, this is a contradiction.