Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 1 / d / Solution

Write $N(w)$ for the inversion set of a Weyl-group element. Part b shows that $s_{\alpha }$ permutes $\Pi \setminus \{\alpha \}$. It follows that right multiplication by $s_{\alpha }$ changes the size of the inversion set byIndeed, all roots other than $\alpha$ are merely relabelled, while $w{s}_{\alpha }(\alpha )=-w(\alpha )$. Part c gives exactly the same recursion for the Coxeter length. Both quantities vanish at the identity, so induction along any word in the simple reflections gives