Conjugacy classes of SL($4,\mathbb{R}$)

Question

I was working on the conjugacy classes of special linear groups of various dimensions. I have succesfully identified the conjugacy classes of SL($2,\mathbb{R}$) by the approach given by Keith Conrad in these notes (see page 6-7).

$\textbf{Primary question:}$

For the next step in my calculations, I need the conjugacy classes of SL($4,\mathbb{R}$), but I cannot find them anywhere. Can anyone help me find them? Ideally, I would like a full derivation but just a statement of what the conjugacy classes are, is also fine! (Or a reference to an article/book where they are given.)

$\textbf{Subsequent question:}$

For now, SL($4,\mathbb{R}$) is all I'm working on, but it is very likely that I'll work on general SL($n,\mathbb{R}$) and SL($n,\mathbb{Z}$), with $n\in\mathbb{N}$, in the future. Does anyone have a reference to an article or book which states conjugacy classes for groups like this?

Thanks in advance for the help!

Answer 1

Consider first the problem of determining when two $n \times n$ square matrices over a field $k$ are conjugate by an element of $GL_n(k)$; working with conjugation by a subgroup of $GL_n(k)$ is a refinement of this so it's good to have a grasp on this case first. If $k$ is algebraically closed the conjugacy classes are labeled by Jordan normal forms up to permutation of blocks, and in general they are labeled by a generalization called rational canonical forms; both of these follow from the structure theorem for finitely generated modules over a PID, since the problem is equivalent to classifying $n$-dimensional modules over $k[x]$ up to isomorphism.

Specialized to $GL_4(\mathbb{R})$, we conclude the following. We have the following cases depending on what the eigenvalues look like:

1. All eigenvalues are real. In this case the rational canonical forms are real Jordan normal forms. Generically these are diagonal matrices, but the following Jordan block structures are also possible: there may be one $2 \times 2$ Jordan block, two $2 \times 2$ Jordan blocks, one $3 \times 3$ Jordan block, or one $4 \times 4$ Jordan block.

2. Two eigenvalues are real and two are complex. In this case the rational canonical forms are direct sums of either a $2 \times 2$ diagonal block or a $2 \times 2$ Jordan block together with a $2 \times 2$ companion matrix of a quadratic polynomial with complex roots.

3. All eigenvalues are complex. In this case the rational canonical forms are either direct sums of two $2 \times 2$ companion matrices or a $4 \times 4$ "Jordan block" of $2 \times 2$ matrices as described in the Wikipedia article on rational canonical forms.

Together with the additional constraint that the product of the eigenvalues is equal to $1$, this answers the question of when two matrices in $SL_4(\mathbb{R})$ are conjugate by an element of $GL_4(\mathbb{R})$. An element of $GL_4(\mathbb{R})$ can be rescaled to have determinant either $1$ or $-1$, so this is almost the same as conjugacy by elements of $SL_4(\mathbb{R})$ except for the additional freedom to conjugate by some, hence any, matrix of determinant $-1$.

This means the classification of conjugacy classes in $SL_4(\mathbb{R})$ involves some of the above cases splitting into two cases; I don't know off the top of my head what that splitting looks like, unfortunately.