I am trying to compute the positive roots of $G_2$ starting from its simple roots. Our instructor has given us an algorithm to do that which goes as follows:
First of all, if $\Sigma=\{\alpha_1,\dots,\alpha_k\}$ are the simple roots and $\alpha=n_1\alpha_1+\dots+n_k\alpha_k$ is a positive root (hence $n_i\geq0$), then the height of $\alpha$ is $m=n_1+\dots+n_k$.
The algorithm goes as follows. If we know the positive roots of height $m$, then we construct the positive roots of height $m+1$ by them: if $\alpha$ is a root of height $m$, we check the inner products $(\alpha,\alpha_i)$. If $(\alpha,\alpha_i)>0$, then $\alpha+\alpha_i$ is not a root. If $(\alpha,\alpha_i)\leq0$ AND $\alpha+\alpha_i$ is not a multiple of another root that we have already obtained, then $\alpha+\alpha_i$ is a positive root. The algorithm terminates when we stop obtaining roots of higher height.
I have used this in many examples and it seems to work just fine. But In $G_2$ it fails! here are my computations, the error appears at height 4. I assume that $\|\alpha_1\|>\|\alpha_2\|$, so $<\alpha_1,\alpha_2>=-3, <\alpha_2,\alpha_1>=-1$, where $<\alpha,\beta>=\frac{2(\alpha,\beta)}{\|\beta\|^2}$.
Height=1: obviously we only have the two simple roots $\alpha_1,\alpha_2$.
Height=2: since $(\alpha_1,\alpha_2)<0$ $\alpha_1+\alpha_2$ is a positive root.
Height=3: The only root of height 2 is $\alpha_1+\alpha_2$. We compute $(\alpha_1+\alpha_2,\alpha_1)$. Since we care only about the sign, we compute $<\alpha_1+\alpha_2,\alpha_1>$ instead. This is linear in the first argument, so $<\alpha_1+\alpha_2,\alpha_1>=<\alpha_1,\alpha_1>+<\alpha_2,\alpha_1>=2-1>0$, hence $2\alpha_1+\alpha_2$ is not a root.
Now we compute $<\alpha_1+\alpha_2,\alpha_2>=<\alpha_1,\alpha_2>+<\alpha_2,\alpha_2>=-3+2<0$, so $\alpha_1+2\alpha_2$ is a positive root.
Height=4: The only root of height 3 is $\alpha_1+2\alpha_2$. We need not compute $<\alpha_1+2\alpha_2,\alpha_1>$, because $(\alpha_1+2\alpha_2)+\alpha_1=2(\alpha_1+\alpha_2)$ can not be a root, as a multiple of another positive root.
So we need only compute $<\alpha_1+2\alpha_2,\alpha_2>=<\alpha_1,\alpha_2>+2<\alpha_2,\alpha_2>=-3+2*2=-3+4=1>0$, so $\alpha_1+3\alpha_2$ is not a root. But this is known to be false! What is wrong here? Is the algorithm not working or am I applying it wrong? Please help me out, this is driving me crazy.
Best Answer
I think you can actually see why things go wrong if you recall the interpretation of the inner products in terms of "$\alpha$-strings through $\beta$". This shows that $(\alpha,\alpha_i)<0$ implies that $\alpha+\alpha_i$ has to be a root (since the string cannot just go to the "negative side". However, $ (\alpha,\alpha_i)>0$ just says that the "negative side" of the string is longer than the "positive side". In most cases, root strings are short enough to ensure that this means that $\alpha+\alpha_i$ is not a root. But in $G_2$ you get a string of length four for which things go wrong. (And I believe that this essentially is the only case where this happens.)