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 to the group — we will often simply write — of invertible complex matrices. The positive integer 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 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 to the matrix . It should be clear that the identity of 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 for the trivial representation we find:
Let’s diverge from the symmetric group for a moment and consider the cyclic group . This consists of powers of a single generator with the relation . That is, we have . The definition of a representation tells us that we must send to , but we may have some latitude in choosing where to send the generator . Since it has to go somewhere, let’s set . And now we know what happens to everything else in the group!
in particular, we have , so these must go to the same value. That is,
and we must have . Thus we have such a representation for each of the th roots of unity, and these are all possible one-dimensional representations of .
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 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
It’s straightforward to check that the additive identity is sent to the identity matrix, and that the group operation is preserved: