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Maximal tori: Overview
Maximal tori: overview
We now want to understand groups more general than $SU(2)$ and $SU(3)$ . One of the key ingredients we used to study those special groups was the weight diagram, which encoded the information about how a representation decomposes under the action of a the diagonal subgroup: $$T=\{\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right):{e}^{i\theta}\in U(1)\}\cong U(1)$$ inside $SU(2)$ and $$T=\{\left(\begin{array}{ccc}\hfill {e}^{i{\theta}_{1}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i{\theta}_{2}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {e}^{i({\theta}_{1}+{\theta}_{2})}\hfill \end{array}\right):({e}^{i{\theta}_{1}},{e}^{i{\theta}_{2}})\in U{(1)}^{2}\}\cong U{(1)}^{2}$$ inside $SU(3)$ .
It's called a torus by analogy: $U(1)\times U(1)$ is topologically a 2dimensional torus (doughnut shape) as a torus has two angular coordinates like $U(1)\times U(1)$ . Higher dimensional tori are defined to be products of more copies of $U(1)$ .
A torus is called
We're focusing on maximal tori because these will give us the most refined decomposition into weight spaces: if you pick a submaximal torus you will always be able to decompose your weight spaces further using the action of the additional $U(1)$ factors of a bigger, maximal torus. For example, if we'd just used ${\theta}_{1}$ in $SU(3)$ , we wouldn't have seen the nice 2dimensional weight diagrams that gave us so much information: we would only have seen projections of these diagrams down to a 1dimensional line.
Examples of maximal tori
For $SU(n)$ , the diagonal subgroup $$T=\{\left(\begin{array}{ccc}\hfill {e}^{i{\theta}_{1}}\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \mathrm{\ddots}\hfill & \hfill \hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill {e}^{i{\theta}_{n}}\hfill \end{array}\right):{\theta}_{1}+\mathrm{\cdots}+{\theta}_{n}=0\}$$ is a maximal torus. This specialises to our earlier examples for $n=2$ and $3$ .
The subgroup $$T=\{\left(\begin{array}{ccc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right):\theta \in \mathbf{R}\}$$ of rotations around the $z$ axis is a maximal torus in $SO(3)$ .
In $SO(4)$ , we can find the 2dimensional maximal torus: $$T=\{\left(\begin{array}{cccc}\hfill \mathrm{cos}{\theta}_{1}\hfill & \hfill \mathrm{sin}{\theta}_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}{\theta}_{1}\hfill & \hfill \mathrm{cos}\theta \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{cos}{\theta}_{2}\hfill & \hfill \mathrm{sin}{\theta}_{2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{sin}{\theta}_{2}\hfill & \hfill \mathrm{cos}{\theta}_{2}\hfill \end{array}\right):{\theta}_{1},{\theta}_{2}\in \mathbf{R}\}$$
In $SO(5)$ we can still only find a 2dimensional maximal torus: $$T=\{\left(\begin{array}{ccccc}\hfill \mathrm{cos}{\theta}_{1}\hfill & \hfill \mathrm{sin}{\theta}_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}{\theta}_{1}\hfill & \hfill \mathrm{cos}\theta \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{cos}{\theta}_{2}\hfill & \hfill \mathrm{sin}{\theta}_{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{sin}{\theta}_{2}\hfill & \hfill \mathrm{cos}{\theta}_{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right):{\theta}_{1},{\theta}_{2}\in \mathbf{R}\}$$ There's no room for another diagonal 2by2 block.
This is one of the first points where we see a significant difference between odddimensional rotation groups and evendimensional rotation groups: relative to the dimension of the group, there's a lot more offdiagonal stuff in odddimensional rotation groups.
Theorem
Let $G$ be a compact, pathconnected matrix group (bounded matrix entries). Then:

$G$ contains a nontrivial torus if $G$ is nontrivial.

Any element of $G$ is contained in a torus.

Any torus is contained in a maximal torus.

Any two maximal torus ${T}_{1}$ and ${T}_{2}$ are conjugate, i.e. there exists an element of $g$ such that ${T}_{2}=g{T}_{1}{g}^{1}$ .
We will not prove (4): the proof would take us too far away from the main themes of the course. The proof uses some very cool ideas from topology, including the Lefschetz fixed point theorem. You can do an indepth project on this, but it does require a bit of background in topology. We will also not prove (2) fully, though we will make a remark about what we would need for this.
Before we move on to the proof, here are some examples to see why the hypotheses of the theorem are necessary.
Suppose $G$ is the real numbers with addition. Since the real line doesn't contain a topologically embedded circle, there's no way to get a nontrivial torus in $G$ . The real line is pathconnected, but not compact, so we see that compactness is necessary.
The assumption of pathconnectedness is necessary for part (2) of the theorem. For example, consider $G=O(2)$ , the group of orthogonal 2by2 matrices. This has two components: the rotations and the reflections. Each component is topologically a circle. The identity component (the subgroup of rotations) is a torus; this the only torus (since any torus is connected and contains the identity) and none of the reflections is contained in it, so the claim that every element of $G$ is contained in a torus fails.
Preclass exercise
Can you see why the tori we have been using in $SU(2)$ and $SU(3)$ are maximal?