How to calculate the reflection of a root system?

Question

We all know that there are four non-isomorphic (finite crystallographic) root systems of rank 2, called $A_1 × A_1$, $A_2$, $B_2$ and $G_2$.For example:

My question is how to calculate the reflection $\sigma_{\alpha_i},i=1,2$?

Answer 1

Geometrically, $\sigma_i$ is the reflection across the hyperplane orthogonal to $\alpha_i$. In $B_2$ above for instance, from the pictures it's geometrically clear that the reflection of $\alpha_2$ across the hyperplane orthogonal to $\alpha_1$ is $2\alpha_1+\alpha_2$, so its coordinate vector relative to the basis $\{\alpha_1,\alpha_2\}$ is $[2,1]^T$, which is precisely the second column of $\sigma_{\alpha_1}$. The columns of all the other matrices above can be checked similarly if you were curious how those matrices were determined.

More generally, the formula for $\sigma_i$ is $$\sigma_i(\lambda)=\lambda-2B(\alpha_i,\lambda)\alpha_i $$ where $B$ is the bilinear form on $V$ satisfying $B(\alpha_s,\alpha_{s'})=-\cos\frac{\pi}{m(s,s')}$, where $m(s,s')$ is specified by the defining relations of the Coxeter group. By computing $B$, you can find the matrix forms of the $\sigma_i$ by computing their action on the desired basis.