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

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.