Some Sample Representations

As promised, we want to see some examples of matrix representations for those who might not have seen much of them before. These are homomorphisms from a group $G$ to the group $GL_d(\mathbb{C})$ — we will often simply write $GL_d$ — of invertible $d\times d$ complex matrices. The positive integer $d$ is called the “dimension” or “degree” of the representation. The first few of our examples will be one-dimensional, largely because that makes the examples simpler. Indeed, a $1\times1$ matrix is simply a complex number, and multiplication of such matrices is just complex multiplication!

First off, every group has the “trivial representation”, which sends each group element $g\in G$ to the matrix $\begin{pmatrix}1\end{pmatrix}$ . It should be clear that the identity of $G$ is sent to the identity matrix — indeed, every group element is sent to the identity matrix — and that the image of the product of two group elements is the product of their images. Writing $\rho_1$ for the trivial representation we find:

$\displaystyle\rho_1(g)\rho_1(h)=\begin{pmatrix}1\end{pmatrix}\begin{pmatrix}1\end{pmatrix}=\begin{pmatrix}1\end{pmatrix}=\rho_1(gh)$

Another one we’ve seen is the signum representation $\mathrm{sgn}$ of a permutation group, which sends even permutations to $\begin{pmatrix}1\end{pmatrix}$ and odd permutations to $\begin{pmatrix}-1\end{pmatrix}$ .

Let’s diverge from the symmetric group for a moment and consider the cyclic group $\mathbb{Z}_n$ . This consists of powers of a single generator $g$ with the relation $g^n=e$ . That is, we have $G=\left\{e,g,g^2,g^3,\dots,g^n=e\right\}$ . The definition of a representation tells us that we must send $e$ to $\begin{pmatrix}1\end{pmatrix}$ , but we may have some latitude in choosing where to send the generator $g$ . Since it has to go somewhere, let’s set $\rho_c(g)=\begin{pmatrix}c\end{pmatrix}$ . And now we know what happens to everything else in the group!

$\displaystyle\rho_c(g^k)=\rho_c(g)^k=\begin{pmatrix}c\end{pmatrix}^k=\begin{pmatrix}c^k\end{pmatrix}$

in particular, we have $g^n=e$ , so these must go to the same value. That is,

$\displaystyle\begin{pmatrix}c^n\end{pmatrix}=\rho_c(g^n)=\rho_c(e)=\begin{pmatrix}1\end{pmatrix}$

and we must have $c^n=1$ . Thus we have such a representation for each of the $n$ th roots of unity, and these are all possible one-dimensional representations of $\mathbb{Z}_n$ .

This illustrates a common technique for finding representations, at least when we have a nice finite presentation of our group: choose an image for each of the generators, and use the relations to give us equations which these images must satisfy. The fact that this works out is deeply wrapped up in the universal properties of free groups, but if that sounds scary you don’t have to worry about it!

We should include at least one example of a higher-degree representation. A nice one is the following representation of the group $\mathbb{Z}$ of integers. Remember that the usual operation on integers is addition, not “multiplication” like we usually say for groups. With that in mind, we set

$\displaystyle\rho(n)=\begin{pmatrix}1&n\\{0}&1\end{pmatrix}$

It’s straightforward to check that the additive identity $0$ is sent to the identity matrix, and that the group operation is preserved:

$\displaystyle\rho(m)\rho(n)=\begin{pmatrix}1&m\\{0}&1\end{pmatrix}\begin{pmatrix}1&n\\{0}&1\end{pmatrix}=\begin{pmatrix}1&m+n\\{0}&1\end{pmatrix}=\rho(m+n)$

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The Unapologetic Mathematician

Mathematics for the interested outsider