# Eigenvectors of action span the representation

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.

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 $$V'$$.