I'm trying to self-teach some Lie theory, and in particular I'm trying to construct the root diagram for $B_2$. I've found 8 roots, labelled $\pi_1,\pi_2,-\pi_1,-\pi_2,\pi_1+\pi_2,-(\pi_1+\pi_2),\pi_1+2\pi_2$ and $-(\pi_1+2\pi_2)$. To draw the root diagram in an $(x,y)$ plane, (and here's where I think I might be going wrong) I've given $\pi_1$ the coordinates $(0,1)$, and thus (via a 135 degree rotation) $\pi_2$ has coordinates $(-\frac{\sqrt2}{2},-\frac{\sqrt2}{2})$. I've then tried to find the coordinates of all the other roots from there. I have two (or maybe more) problems:
- The coordinates for $\pi_2$ that result from the 135 degree rotation are inconsistent with the notion that one fundamental root has to be longer than the other (I've chosen $\pi_1>\pi_2$).
- The root diagram that results from the process described above is clearly incorrect - the diagrams I've found online/in textbooks show pairs of roots that are 90 degrees from eachother, which mine does not.
If anyone could shed any light on where I'm going wrong, it'd be much appreciated.
Since you know that the two simple roots have different length, you cannot get from on to the other by a rotation. So $\pi_2$ should be a positive multiple of the vector you have. To choose the right factor you can either use what you know about the length ratio of the two roots or what you know about their inner product.Anyway the result should be $(-\frac12,\frac12)$. This leads to the familiar picture (maybe roatated because of the choice where the long root points).