The big picture is that I am trying to understand how the monodromy group of functions defined on $\mathbb{C} \setminus \{z_1, z_2, z_3,\ldots, z_n\}$ is related to the fundamental group (first homotopic group) of the set itself, i.e. $\pi_1(\mathbb{C} \setminus \{z_1, z_2, z_3,\ldots,z_n \})$. I understand the latter very well, $\pi_1(\mathbb{C} \setminus \{z_1, z_2, z_3,\ldots,z_n \}) \cong F_n$, where $F_n$ denotes the free group on $n$ generators. But I do not understand the former very well.
This background aside, my basic question is quite self-contained:
On the wikipedia article for group representations, in the "Examples" section, they give this example:
Consider the complex number $u = e^{2\pi i / 3}$ which has the
property $u^3 = 1$. The set $C_3 = \{1, u, u^2\}$ forms a cyclic group
under multiplication. This group has a representation $\rho$ on
$\mathbb{C}^2$ given by:$$ \rho(1) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad
\rho(u) = \begin{bmatrix} 1 & 0 \\ 0 & u \end{bmatrix}, \quad
\rho(u^2) = \begin{bmatrix} 1 & 0 \\ 0 & u^2 \end{bmatrix}. $$This representation is faithful because $\rho$ is a one-to-one map.
They proceed to give some further "group representations" in $\mathbb{C}^2$ and $\mathbb{R}^2$. What confuses me, is this seems to be nothing other than a group isomorphism. It seems like they have simply given an example to support the claim that: "Given any group $G$, there exists a set of matrices in $\operatorname{GL}(n,\mathbb{R})$ such that a group of those matrices under the operation of matrix multiplication is isomorphic to $G$". If I remember correctly this claim happens to be true for finite groups only. This may be related to the fact that the action is faithful, but they don't give an example of a non-faithful one.
Regardless, I don't see any group action going on here, I don't seem to be acting with the elements of $\{1, u, u^2\}$ on any set? I thought the whole point of a group representation was that it was a group action on a vector space? Where are the vectors?
What am I missing?
Thanks!
Best Answer
There are two ways one can define a representation of a group $G$ on a vector space $\mathbb{C}^n$ (The field does not have to be $\mathbb{C}$, nor does one need to specify a collection of 'standard' basis like I have done, but it just makes everything simpler).
Definition 1: A representation of a group $G$ on $\mathbb{C}^n$ is a group homomorphism: $$\rho:G\rightarrow \operatorname{GL}_n(\mathbb{C}).$$
Definition 2: A representation of a group $G$ on $\mathbb{C}^n$ is a group action by $\mathbb{C}$-linear homomorphisms: $$\rho:G\times\mathbb{C}^n\rightarrow \mathbb{C}^n.$$
I shall leave it to you to show that these two are equivalent definitions in a very 'natrual' way.
Now, one says that a representation is faithful if:
Again, I shall leave it to you to show that these two definitions of faithful actions are also the same in very 'natural' way!
For any group, a simple example of a non-faithful action is the homomorphism $G\rightarrow \operatorname{GL}_n(\mathbb{C})$ which take every element to the identity matrix. And, in your specific example where $G=C_3$, every representation is either faithful or trivial (i.e. everything goes to identity).
Next, your claim that the one can get a faithful action only when the group is finite is not true. For example, the representation: $$\mathbb{C}\rightarrow \operatorname{GL}_2(\mathbb{C})$$ which takes a $z\in\mathbb{C}$ to the matrix $\begin{bmatrix} 1 & z \newline 0 & 1\end{bmatrix}$ is a faithful representation (since you put algebraic geometry in your tags, maybe I should add that this generalizes to a representation of algebraic groups - if this statement makes no sense to you, you can completely ignore it).