Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 1

A root system in the real inner product space $V$ is a finite spanning set $\Phi \subset V\setminus \{0\}$ such thatwhere the orthogonal reflectionFor a crystallographic root system one additionally requires $2(\beta ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$; that condition is not needed for general finite reflection groups.

A fundamental system of a root system is a basis $\Delta \subset \Phi$ such that every root has either all nonnegative or all nonpositive coordinates in this basis. Its associated positive system of a root system isand $\Phi =\Pi \bigsqcup (-\Pi )$.

Solved by gpt-5.6-sol high.

The reflection group of a root system isEvery generating reflection permutes $\Phi$, so every $w\in W(\Phi )$ does too. If $\Delta$ is fundamental, then $w\Delta$ is a basis contained in $\Phi$. Writingshows thatthe coefficients are unchanged and therefore still have one sign. Thus $w\Delta$ is another fundamental system, and $W(\Phi )$ acts on the set of all fundamental systems.

Solved by gpt-5.6-sol high.

Use the positive-root criterion for Coxeter length: for a simple root $\alpha \in \Delta$,The hypothesis $w\Delta \subseteq \Pi$ therefore says that every simple generator is a right ascent of $w$. If $w\ne 1$, a reduced expression in a Coxeter group for $w$ has a final simple generator $s_{\alpha }$, and deleting it givesa contradiction. Hence $w=1$.

If $w$ stabilizes $\Delta$ setwise, then $w\Delta =\Delta \subseteq \Pi$, so the result just proved gives $w=1$. Thus the stabilizer of every fundamental system is trivial.

Solved by gpt-5.6-sol high.

If a finitely generated Coxeter group $W$ is finite, its integer-valued Coxeter length has a maximum. Conversely, if some $w_0$ has globally maximal length $N$, every group element has a word of length at most $N$. There are only finitely many words of bounded length in the finite set of simple generators, so $W$ is finite.

Realize the finite group as the reflection group of a root system with fundamental system $\Delta$ and positive system $\Pi$. Maximality and the fact that multiplication by a simple generator changes Coxeter length by one giveThe positive-root criterion for Coxeter length therefore gives $w_0\Delta \subseteq -\Pi$. Since $w_0\Delta$ is itself fundamental, it must be the simple system $-\Delta$ of the positive system $-\Pi$.

If $v_0$ is another maximal-length element, the same argument gives $v_0\Delta =-\Delta =w_0\Delta$. Hence $v_0^{-1}{w}_0$ stabilizes $\Delta$, and part c gives $v_0^{-1}{w}_0=1$. The Longest element of a finite Coxeter group is therefore unique.

Solved by gpt-5.6-sol high.