The <reflection group of a root system> is
$$
W(\Phi)=\langle s_\alpha:\alpha\in\Phi\rangle\leq O(V).
$$
Every generating reflection permutes $\Phi$, so every $w\in W(\Phi)$ does too. If $\Delta$ is fundamental, then $w\Delta$ is a basis contained in $\Phi$. Writing
$$
\beta=\sum_{\alpha\in\Delta}c_\alpha\alpha
$$
shows that
$$
w\beta=\sum_{\alpha\in\Delta}c_\alpha w\alpha;
$$
the 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-111/1/b/solution