# Dihedral angles

## Dihedral angles

### Statement of result

Lemma:

The dihedral angle between two of the root hyperplanes in a root system can be $90$ , $60$ , $45$ or $30$ degrees and nothing else.

### Explanation

By the angle between two hyperplanes, I mean the following:

• take the unit vectors $v$ and $w$ orthogonal to these hyperplanes,

• take the smallest angle $\phi$ such that $\cos\phi=v\cdot w$ .

In a root system, the vectors orthogonal to the root hyperplanes are the (simple) roots, so we are really interested in $\alpha\cdot\beta$ where $\alpha$ and $\beta$ are roots. Recall that the dot product is the Killing form.

### Proof

Since we're in a root system, if $\alpha$ and $\beta$ are roots then $P_{\alpha}(\beta)=\frac{1}{2}n_{\beta\alpha}\alpha$ for some $n_{\beta\alpha}\in\mathbf{Z}$ , where $P_{\alpha}(\beta)$ is the orthogonal projection of $\beta$ onto the line through $\alpha$ .

Example:

In the $SU(3)$ root system shown below, the projection of $\beta$ to the line through $\alpha$ gives $-\alpha/2$ .

By Euclidean geometry, we have $P_{\alpha}(\beta)=\frac{\alpha\cdot\beta}{\alpha\cdot\alpha}\alpha.$

This means that $n_{\beta\alpha}=\frac{2\alpha\cdot\beta}{\alpha\cdot\alpha}.$ We also have $n_{\alpha\beta}=\frac{2\alpha\cdot\beta}{\beta\cdot\beta},$ so $n_{\beta\alpha}n_{\alpha\beta}=\frac{4(\alpha\cdot\beta)^{2}}{(\alpha\cdot% \alpha)(\beta\cdot\beta)}=4\cos^{2}(\phi),$

where $\phi$ is the angle between the roots.

We have $-1<\cos\phi<1$ because the roots are not collinear, so $n_{\alpha\beta}n_{\beta\alpha}=4\cos^{2}\phi\in\{0,1,2,3\}$ because it's an integer between 0 and 4.

These possibilities give $\phi=\begin{cases}90\\ 60\\ 45\\ 30\end{cases}$ degrees.