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

The crystallographic axiom makesintegers. If $\theta$ is the angle between the roots, thenThis is a nonnegative integer. Since $\beta \ne \pm \alpha$, the roots are not parallel, so ${\cos }^2\theta <1$. Thereforewhich is the root-system finiteness lemma.

Choose simple roots $\alpha ,\beta$. Their inner product is nonpositive, so their angle lies in $[\pi /2,\pi )$. Part i leaves four possibilities for $mn$:

The root strings generated by the two simple reflections produce exactly the roots in those four standard systems. Hence these are all possibilities, proving the classification of rank-two root systems.

An irreducible root system is simply laced when every root has the same length, equivalently when its Dynkin diagram has no multiple edge.

If all roots have the same length, the two Cartan integers for $\alpha$ and $\beta$ are equal. The root-system finiteness lemma then makes their product either zero or one, sofor $\alpha \ne \pm \beta$.

Conversely, when all such Cartan integers lie in $\{0,\pm 1\}$, any two nonorthogonal roots have Cartan integers of absolute value one in both directions. Their squared lengths are therefore equal. Irreducibility makes the graph joining nonorthogonal roots connected, so all roots have the same length. Thus the system is simply laced.

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.