# How to prove that $\sum_{g \in G} gv=0$ on an irreducible representation?

I want to prove that $\sum_{g \in G} gv=0$ on an irreducible representation $p: G \to GL(V)$ with $dim_{\mathbb{C}}(V) \geq 2$. In additional if i have an $h \in G$ and $gh=hg \ \ \ \forall g \in G$, is it possible to find that $p(h)= \zeta I$ for a root of unity $\zeta \in \mathbb{C}$ using the first part of my question?

Using Schur's Lemma i can find that for each $c \in Z(G)$ there exist a $z \in \mathbb{C}$ such that $p(c)v= zv$ but I'm completely lost.

## 1 Answer

Suppose that $\sum_{g \in G} gv \neq 0$. Then the span of this vector is $G$-stable, not $0$, and not all of $V$ (since the dimension of $V$ is greater than $1$).