What is the difference between a group representation and an isomorphism to GL(n,R)?

Question

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!

Answer 1

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:

• In Definition 1: the map $\rho$ is injective;
• In Definition 2: if the group action is faithful.

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).

Answer 2

Let $G$ be a finite group and $V$ a finite-dimensional complex vector space. Then a representation of $G$ on $V$ is a group homomorphism $\rho: G \to \operatorname{GL}(V)$. (Note that $\rho$ can never be surjective and is not required to be injective.) Then, the elements of $G$ act on $V$ via $g \cdot v = \rho(g)(v)$. (Note that this action is not necessarily faithful.)

In this example, we have $G = \{1, u, u^2\} \subseteq \mathbb{C}^*$, $V = \mathbb{C}^2$ and $$ \begin{array}{c} 1 \cdot \begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix} \\ u \cdot \begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix} = \begin{bmatrix} v_1 \\ u v_2 \\ \end{bmatrix} \\ u^2 \cdot \begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix} = \begin{bmatrix} v_1 \\ u^2 v_2 \\ \end{bmatrix} \\ \end{array} $$

Answer 3

I would say that the most common definition of a (linear) representation of a finite group $G$ on a finite-dimensional complex vector space $V$ is that it is a homomorphism $\rho:G\to\DeclareMathOperator{\GL}{GL}\GL(V)$. However, there are several reformulations of this definition.

To begin with, recall that a finite-dimensional vector space is classified up to isomorphism by its dimension. Thus, in our attempts to classify the representations of $G$, it usually suffices to only consider representations taking the form $\rho:G\to\GL(\mathbb C^n)$. Furthermore, there is a canonical group isomorphism $\GL(\mathbb C^n)\to\GL_n(\mathbb C)$ given by identifying each invertible linear map $\varphi:\mathbb C^n\to\mathbb C^n$ with the matrix that represents it under the standard basis of $\mathbb C^n$. Thus, we often speak of "representations $G\to\GL(n,\mathbb C)$" – more properly, these could be called matrix representations.

There is also a natural correspondence between representations of a group $G$ on $V$ and "linear actions" of $G$. These are group actions $G\times V\to V$ that respect the structure of $V$ as a vector space – specifically, a group action $G\times V\to V$ is a linear action if $g\cdot (v+w)=(g\cdot v)+(g\cdot w)$ and $g\cdot(\lambda v)=\lambda(g\cdot v)$. Given a linear action of $G$ on $V$, we obtain a map $\rho:G\to V^V$ by setting $\rho(g)(v)=g\cdot v$. It is routine to check that $\rho(g)$ is a linear automorphism of $V$ for all $g\in G$, and so restricting the codomain of $\rho$ yields a representation of $G$. Similarly, each representation $\rho:G\to\GL(V)$ induces a linear action of $G$ on $V$, and these processes are mutual inverses of each other. The passage from a representation of a group to its corresponding linear action is an instance of currying.

(All of these definitions and basic theorems work just as well if we allow $G$ to be an infinite group, or $V$ to be an arbitary vector space over a field $k$. However, the representation theory of groups is much less well understood in the situation where $G$ is infinite – and even in the case where $G$ is finite, things are easiest if $k$ is algebraically closed and of characteristic zero, and $V$ is finite-dimensional.)