In the Fulton/Harris book "Representation theory: a first course", section 1.3:
We are looking at any representation W of $S_3$, by just looking at the abelian group $U_3 \sim Z/3Z \subset S_3$. Let $\tau$ be any generator of $U_3$.
Why is that the book says "the space W is spanned by the eigenvectors of the action $\tau$", and "whose eigenvalues are of cause all the powers of $\omega = e^{2\pi i/3}$? Can someone elaborate on this?