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

On $P=\mathrm{span}\{\alpha ,\beta \}$, each $s_{\alpha }$ is reflection in the line ${\alpha }^{\perp }\cap P$. The product of two plane reflections is a rotation. If $\theta$ is the oriented angle from $\beta$ to $\alpha$, thenrotates through $2\theta$ modulo $2\pi$.

If this rotation has order $m$, conjugation by either reflection inverts it. Henceis a dihedral group, with rotational subgroup $⟨r⟩=⟨s_{\alpha }{s}_{\beta }⟩$.

For the simple-root angles from part ii, the rotation orders and Weyl groups areThese are the rank-two Weyl groups.