## Conjugacy classes in the finite general and special linear groups

Now that I’m finally done with school for the summer, I’d like to get back into the routine of blogging regularly. If you were following last summer: I never completed my project of understanding the Weil representation, so I probably won’t be continuing that series of posts. I may be helping some people complete that project this summer, in which case I can hopefully link to some further information eventually.

This week I’m going to give a detailed description of the conjugacy classes in and , where is the finite field with elements. This is relevant to representation theory because the conjugacy classes in a finite group correspond bijectively to irreducible representations, and in particular we will find out how many irreducible representations these groups have. A quick Google search reveals that it is easy to find the final answers, but somewhat harder to find a careful explanation, which is what I will attempt now.

First, the general linear group: for any , consider the -module where acts by . Two matrices are conjugate if and only if the corresponding modules are isomorphic, and it is easy to analyze these isomorphism classes using the structure theorem for principal ideal domains. Note that since we are counting invertible matrices, we need only consider polynomials with nonzero constant term.

- The nonzero scalar matrices are precisely the center of , so these account for conjugacy classes with one element each.
- For each such that , there is the semisimple conjugacy class of matrices with minimal polynomial : the centralizer of such a matrix is a split maximal torus , so each of these conjugacy classes has elements.
- For each there is a conjugacy class of matrices with minimal polynomial which are not semisimple, and hence conjugate to a Jordan block. If we write a Jordan block as , where is the nilpotent matrix defined by and , it is easy to see that the centralizer consists of matrices of the form where and . Thus each of these conjugacy classes has elements.
- Finally, there are the matrices which have no eigenvalue in , and therefore have a conjugate pair of eigenvalues . Such matrices are semisimple because is perfect, so their conjugacy class is determined by their eigenvalues, and in particular we see that there are conjugacy classes of these matrices. If has eigenvalue , then the subalgebra is isomorphic to . If we use the basis to identify with , we get an isomorphism , and here corresponds to . The centralizer of this subalgebra is , so we see that the centralizer of in is isomorphic to the non-split torus and in particular the conjugacy class of has elements.

Note that the total number of conjugacy classes of is

As for , we first find the -conjugacy classes in and then determine how they split into -conjugacy classes. Unfortunately, we must now keep track of whether is even or odd.

- The center of is trivial if is even or if is odd. Hence this accounts for one conjugacy class if is even or two if is odd, with one element each in either case.
- For each with , there is the semisimple conjugacy class of matrices with minimal polynomial . If is even then there are of these conjugacy classes, and if is odd then there are . We already saw that the stabilizer of such a matrix in is a split maximal torus, so each conjugacy class has elements.
- There are matrices with minimal polynomial which are not semisimple, and hence conjugate to a Jordan block. If is even then there is only one such-conjugacy class, and if is odd then there are two. We saw that the stabilizer in of such a matrix has elements, so these -conjugacy classes contain matrices each.
- The conjugacy classes of matrices which have no eigenvalue in are parameterized by conjugate pairs where . The latter equation has solutions in the cyclic group , and if is even only one of those solutions comes from , while if is odd then two do. Thus there are such conjugacy classes if is even and if is odd. As we saw, the stabilizers in of these matrices are non-split maximal tori, so each of these conjugacy classes has elements.

So we have described the -conjugacy classes in , but it remains to see how these split as -conjugacy classes. We will show momentarily that semisimple -conjugacy classes in do not split further as -conjugacy classes, and here the only non-semisimple matrices are conjugate to one of the Jordan blocks (where is the nilpotent matrix mentioned earlier). Let’s write and for the moment to improve the notation. Now if is the -conjugacy class in of , then as -sets and in particular as -sets. In particular we get a bijection . We saw earlier that if is a Jordan block then consists of matrices of the form with and , so is the subgroup of squares. Thus if is odd then the two -conjugacy classes of split into two -conjugacy classes with elements each, and if is even then the -conjugacy class of does not split further as an -conjugacy class. We see now that if is odd then has

conjugacy classes, and if is even then the number is

It remains to show that if is a semisimple -conjugacy class, then does not split further as an -conjugacy class. This is true for and where is arbitrary and is any field with the property that the norm map is surjective for any finite extension . Even more generally, suppose is a finite-dimensional commutative semisimple algebra over such a field , and a finite-dimensional -module. Then we have the determinant map , and we claim the subgroup surjects onto . Now by the semisimplicity hypothesis, where each is a finite field extension of , so where each is an -vector space and acts diagonally. Thus the automorphism group splits as well:

.

It is enough to show that is surjective for some , so we have reduced to the case that is a finite field extension of . But now we can see from the definitions that the determinant factors into the determinant followed by the norm , and the latter is surjective by assumption. Applying this to the case when is semisimple, , and , we have and the claim follows.