I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I am trying to calculate the angles in a triangle.
Given a root system $\Phi$ with simple system $\Delta$,
$$H_{\alpha} = \{\lambda \in V : \langle \lambda, \alpha \rangle = 0 \}$$ are the hyperplanes and the connected components of
$$V \setminus \bigcup_{\alpha} H_{\alpha}$$ are the chambers. Given a chamber $C$ corresponding to a simple system $\Delta$, the walls are defined to be the hyperplanes $H_{\alpha}$ for $\alpha \in \Delta$.
So, from what I gather, to find the angles of the fundamental alcove/chamber I need to compute the angle between these hyperplanes.
So, for the $B_2$ case, I believe the simple roots are $e_1 - e_2$ and $e_2$. However, how can there be a triangle formed by two vectors? Even ignoring this, the angle between these two vectors doesn't come out to the right answer. For the $B_2$ case, the answer is supposed to be $\pi/2, \pi/4, \pi/4$. And, unless I made a mistake in my calculation, the angle between $e_1-e_2$ and $e_2$ is not any of these angles.
What am I doing wrong? What am I misunderstanding?
Not having see your calculation, I don't know where you have gone wrong, but one thing to make sure of i that at every step you are using the bilinear form for the Coxeter system, and not the standard Euclidean inner product. In particular in the definition of the orthogonal hyperplane and dihedral angles.
– David Sheard Apr 26 '22 at 08:56