Eigenvectors of action span the representation

Question

Let $V$ be a complex finite dimensional vector space and $\rho:S_3\to \text{GL}(V)$ be a representation of the symmetric group $S_3$. Let $A_3$ be the alternating subgroup of $S_3$; let $\tau$ be a generator of $A_3$ (since $A_3$ is cyclic.). I want to prove that $V$ is spanned by the eigenvectors of the action of $\tau$ on $V$, and that their eigenvalues are powers of $\omega=\exp(2\pi i/3)$.

I have no idea of how to do this. I think it has to do with the fact that $A_3$ is Abelian, and thus the actions are G-module homomorphisms. Then some application of Schur's Lemma might follow, but I am not sure of how to proceed.

Answer 1

Recursive proof on $dim(V)$, since $S^3$ is finite, there exists an Hermitian product of $V$ invariant by the action of $V$. Let $c$ be an eigenvalue of $\tau$, and $V_c$ the associate eigenspace, $V'$ the orthogonal of $V_c$ is invariant by $\tau$. If $V_c\neq V$, apply the recursive hypothesis to $V'$.