Let $V$ be the real <vector space> with basis $(e_i)_{i\in I}$. The <Coxeter Gram matrix> defines the symmetric <bilinear form>
$$
\langle e_i,e_j\rangle=G(W)_{ij}=-2\cos\left(\frac{\pi}{m_{ij}}\right).
$$
Its diagonal entries are $2$. The <Geometric representation of a Coxeter group> is generated by the reflections
$$
\sigma(x_i)(v)=v-\langle v,e_i\rangle e_i.
$$
Each has square one, and on $\operatorname{span}\{e_i,e_j\}$ the product $\sigma(x_i)\sigma(x_j)$ has order $m_{ij}$. The reflections therefore satisfy the Coxeter relations and define a <group representation> $\sigma:W\to GL(V)$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-111/2/a/solution