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

A fundamental system of a root system is a subset $\Delta \subset \Phi$ which is a basis of $V$ and for which every $\beta \in \Phi$ has an expansionwhose coefficients are either all nonnegative or all nonpositive. Its associated positive system of a root system isThus $\Phi =\Pi \bigsqcup (-\Pi )$, and the elements of $\Delta$ are the simple roots.

Solved by gpt-5.6-sol high.

The Weyl reflection in $\alpha$ satisfiesNow take $\beta \in \Pi \setminus \{\alpha \}$ and expand it in the basis $\Delta$. At least one coefficient belonging to a simple root other than $\alpha$ is positive. Sincethe reflection changes only the coefficient of $\alpha$. Every root has simple-root coefficients of one sign, so the unchanged positive coefficient prevents $s_{\alpha }(\beta )$ from being negative. Hence $s_{\alpha }(\beta )\in \Pi$. Because $s_{\alpha }$ is an involution, it permutes $\Pi \setminus \{\alpha \}$, while it exchanges $\alpha$ and $-\alpha$. Therefore

Solved by gpt-5.6-sol high.

Let $C$ be the fundamental chamber of a root system. The chambers $wC$ and $w{s}_{\alpha }C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha )$. The chamber $wC$ lies on the side on which $w(\alpha )$ is positive. If $w(\alpha )\in \Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha )\in -\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the Coxeter length $\ell (w)$, and adjacent chamber distances differ by one. Consequently

Solved by gpt-5.6-sol high.

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

Solved by gpt-5.6-sol high.

By the assumed transitivity on fundamental systems, some $w_0\in W$ sends $\Delta$ to $-\Delta$. It therefore sends the entire positive system of a root system $\Pi$ to $-\Pi$. Part d then givesFor every $w\in W$, its inversion set is contained in $\Pi$, so $\ell (w)\le |\Pi |$ and $w_0$ has maximal length.

If $u$ also has maximal length, then $N(u)=\Pi$, so $u(\Pi )=-\Pi$. Hence $w_0^{-1}u$ preserves $\Pi$ and has no inversions. Part d makes its Coxeter length zero, so it is the identity. Thus $u=w_0$, proving that the Longest element of a finite Coxeter group is unique and has length $|\Phi |/2$.

Solved by gpt-5.6-sol high.