The crystallographic axiom makes
$$
m=\langle\alpha,\beta^\vee\rangle,
\qquad
n=\langle\beta,\alpha^\vee\rangle
$$
integers. If $\theta$ is the angle between the roots, then
$$
mn=
\frac{2(\alpha,\beta)}{(\beta,\beta)}
\frac{2(\beta,\alpha)}{(\alpha,\alpha)}
=4\cos^2\theta.
$$
This is a nonnegative integer. Since $\beta\ne\pm\alpha$, the roots are not parallel, so $\cos^2\theta<1$. Therefore
$$
\boxed{mn\in\{0,1,2,3\}},
$$
which is the <root-system finiteness lemma>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-102/4/i/solution