= Solution
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.
Back to article page
