Claims on Root systems

Question

For a root system R prove or disprove:

a. Assume that the angle θ between the roots α and β is obtuse (θ > π/2) Then α+β ∈R.

b. The angle θ between α and β is π/2 . Then α+β is not a root.

c. If the roots α and β have the same length then θ = π/3 or 2π/3 .

Using the definition 8.1 in Brian.C Hall's book on lie group, lie algebras and representations, and looking at propositions 8.6 and 8.7 (section 8).

Proposition 8.6. Shows part c (though I did not notice the "if" part explicitly and in my question I do not assume that $(\alpha,\alpha)\geq (\beta,\beta)$).

In the case where $m_1=m_2=1$ they mention that if $(\alpha,\beta) > 0$ then $\theta=\pi/3$ and in the case $m_1=m_2=-1$ if $(\alpha,\beta) < 0$ then $\theta=2\pi/3$. Why this holds, i.e. why we get these angles?

Is it because $(\alpha,\beta)=\sqrt{(\alpha,\alpha)}\sqrt{(\beta,\beta)}cos(\theta)$ ; $0\leq \theta\leq \pi$.

So $(\alpha,\beta)>0$ iff $cos(\theta)>0$ iff $0<\theta<\pi/2$ And $(\alpha,\beta)<0$ iff $cos(\theta)<0$ iff $\pi/2<\theta<\pi$ ?

Proposition 8.7. shows part a. How do we see that ,"the projection of β onto α equals $-\alpha/2$ thus $s_{\alpha}\beta=\alpha+\beta$ is again a root"?

Part b should be seen by definition I think. If the angle between the two roots is $\pi/2$ then they are orthogonal and $(\alpha,\beta)=0$, but I did not understand how to conclude that $\alpha+\beta$ is not in $R$.

I would be glad for clarifying this for me.

Answer 1

Part a. From the list in Dietrich Burde's answer to https://math.stackexchange.com/a/1545723/96384, and w.l.o.g. assuming $(\beta, \beta) \ge (\alpha, \alpha)$, infer that when $(\alpha, \beta ) <0$ (which indeed by the cosine definition means $\pi/2 < \theta < \pi$), then either

$(\alpha, \beta)/(\alpha, \alpha) =-1/2$ and $(\beta, \beta) = (\alpha, \alpha)$ and $\theta = 2\pi/3$ (case of $A_2$); or

$(\alpha, \beta)/(\alpha, \alpha) =-1$ and $(\beta, \beta) = 2(\alpha, \alpha)$ and $\theta= 3\pi/4$ (case of $B_2$); or

$(\alpha, \beta)/(\alpha, \alpha) =-3/2$ and $(\beta, \beta) = 3(\alpha, \alpha)$ and $\theta= 5\pi/6$ (case of $G_2$).

Note that in any Euclidean vector space, the projection of $v$ to $w$ is $\dfrac{(v,w)}{(w,w)} \cdot w$. So e.g. the first case says the projection of $\beta$ onto $\alpha$ is $-\alpha/2$.

So in your source, they apparently assume they are in the first case, and since you say you know the definition of the reflections $s_\alpha$, it is immediate in that case that $s_\alpha(\beta) =s_\beta(\alpha) =\alpha +\beta$. Note that in the other two cases, we still have $s_\beta(\alpha) = \alpha +\beta$, while $s_\alpha(\beta) = \beta + 2\alpha$ and $\beta +3\alpha$, respectively.

Part b. Look at the above root system $B_2$ and consider (not the ones called $\alpha, \beta$ above, but) two orthogonal roots.

Part c. Again, look at that root system $B_2$. (Note that if one excludes this counterexample case of $\theta=\pi/2$, the assertion is true, and you proved half of it if you inferred what was stated in a. The case of acute angles is rather analogous.)