# Simple Roots of E6 in Coordinates?

There are several possibilities how one can write simple roots in terms of coordinates. Firstly, they depend on the numbering of the nodes in the Dynkin diagram. Let's fix the choice for $E_6$ to be

Unfortunately there are still several possibilities. For example, this paper quotes at the second page very different simple roots than Wikipedia, which claims that the simple roots in coordinates are the rows of the following matrix:

Which set is correct? Or are both? My problem is that I get very different result if I use these two different sets. Is there some good resource that lists the different possibilities?

• If you apply any rotation to those roots you get another set of vectors that also generates the root system $E_6$. Alternatively you can think of these are representing those vectors using another orthonormal basis. Any set of vectors with the correct pairwise inner products will do as well. So there are infinitely many sets of simple roots of a root system of type $E_6$. There is no way to get rid of this freedom. You need to live with it. Commented Jul 8, 2015 at 13:43
• Anyway, I didn't check but both are probably correct. Commented Jul 8, 2015 at 13:43
• OT as it origins from WP, but: What is the purpose of the colour coding in the Dynken diagram? Commented Jul 8, 2015 at 13:47
• @JyrkiLahtonen Thanks a lot for the clarification. My problem is that I use a software package called LieArt (arxiv.org/pdf/1206.6379v2.pdf), which computes weight systems for arbitrary representations. The software displays the weight in the simple root basis (alpha-basis called there). Therefore, the weights written like this must depend on choice of the simple roots, i.e. must change if we perfom a rotation on the simple root basis?! For different simple root choices, the coefficents w.r.t this basis are different. Unfortunately I can't figure out which basis they use.
– jak
Commented Jul 8, 2015 at 13:54
• That is a non-problem. If you change the basis of the ambient Euclidean space, and consequently the roots become different vectors in $\Bbb{R}^6$, then the weights are changed accordingly. The relation between the roots and the weights does not change. For example with type $A_2$ the fundamental weigts are always $(2\alpha_1+\alpha_2)/3$ and $(\alpha_1+2\alpha_2)/3$. Irrespective of which two plane vectors the simple roots happen to be. Commented Jul 8, 2015 at 13:57

It is straightforward to verify $\langle v,v\rangle = 2$ for all rows $v$ and $\langle v,w\rangle=-1$ precisely for rows joined by a line and $\langle v,w\rangle=0$ otherwise.