Question about a passage in Fulton and Harris So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$.  I came along this passage and had some questions  
What do they mean when they say "the space W is spanned by eigenvectors $v_i$ for the action of $\tau$"  How are they getting that?
 A: Since $\tau$ is of order 3, the minimal polynomial of the action $T \colon W \to W$ of $\tau$ is a factor of $x^3 - 1 = (x - 1)(x - \omega)(x - \omega^2)$. Hence the minimal polynomial of $T$ has no multiple root and therefore $W$ is a semisimple $\mathbb{C}[T]$-module, which means that $W$ is spanned by its eigenvectors (Linear algebra!).
A: It seems to me that calling $W$ the representation is somewhat misleading: it ought to be called the representation space. The representation (where the images of the elements of $S_3$ live) is a subgroup of $GL(W)$.
It might be that $W = \Bbb C^3$, for example, with:
$\tau = (1\ 2\ 3) \mapsto \begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix} = T$
Note that: $\det(xI - T) = \begin{vmatrix}x&0&-1\\-1&x&0\\0&-1&x\end{vmatrix} = x^3 - 1$
so that the eigenvalues of $T$ are: $\lambda_1 = 1, \lambda_2 = -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \lambda_3 = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$.
An eigenvector $v_1$ corresponding to $\lambda_1$ is $v_1 = (1,1,1)$.
An eigenvector $v_2$ corresponding to $\lambda_2$ is $v_2 = (1, -\frac{1}{2} + i\frac{\sqrt{3}}{2}, -\frac{1}{2} - i\frac{\sqrt{3}}{2})$.
An eigenvector $v_3$ corresponding to $\lambda_3$ is $v_2 = (1, -\frac{1}{2} - i\frac{\sqrt{3}}{2}, -\frac{1}{2} + i\frac{\sqrt{3}}{2})$.
In this particular case, $\{v_1,v_2,v_3\}$ form a basis for $\Bbb C^3$, which is certainly a spanning set.

A second example: let $W = \Bbb C^2$ with $\tau \mapsto \begin{bmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix}=R$.
Here, $\det(xI - R) = x^2 + x + 1$, and the eigenvalues are the same as $\lambda_2,\lambda_3$ above. 
An eigenvector corresponding to $\lambda_2$ is $w_1 = (1,-i)$, and an eigenvector corresponding to $\lambda_3$ is $w_2 = (1,i)$. Again, these form a basis for $\Bbb C^2$.

A third, and somewhat less enlightening example:
let $W = \Bbb C$, with $\tau \mapsto 1$. This (the complex number $1$) is a basis for $\Bbb C$.
