The sum of an Irreducible Representation I was hoping someone could help me with the following question.
Let $\rho$ be an irreducible presentation of a finite group $G.$  Prove
\begin{equation*}
\sum_{g \in G} \rho(g) = 0 
\end{equation*}
unless $\rho$ is the trivial representation of degree $1$.
I think I have to use Schur's Lemma which states the following. Let $\rho: G \longrightarrow GL(n,\mathbb{C})$ be a representation of G. Then $\rho$ is irreducible if and only if every $n \times n$ matrix $A$ which satisfies 
\begin{equation*}
\rho(g)A = A\rho(g) \ \ \ \forall \ g \in G
\end{equation*}
has the form $A = \lambda I_n \, $ with $\lambda \in \mathbb{C}$. 
But I am really not sure how the lemma can be applied to this question?

Let $t=\sum_{g\in G}\rho(g)$, which is an linear endomorphism of $V$. The subset $t(V)$ of $V$ is a $G$-submodule of $V$, as you can easily check. Moreover, $G$ acts trivially on all elements of $t(V)$.
If $V$ is irreducible, then either $t(V)=0$ or $t(V)=V$. In the first case, we have that in fact $t=0$. In the second one, we see that $G$ acts trivially on all of $V$, so $V$ must be of dimension $1$.
