Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 4 / i / Solution

In the crystallographic convention for this course, an admissible Coxeter graph comes from linearly independent unit vectors $v_1,\dots ,v_r$ in a Euclidean space, such that for $i\ne j$,The vertices $i,j$ have no connection when $n_{ij}=0$, and a single, double or triple connection when $n_{ij}=1,2,3$. Equivalently, the symmetric Gram matrixmust be positive-definite. This equivalence follows because a positive definite Gram matrix can be realized by linearly independent vectors, while such vectors necessarily have a positive definite Gram matrix. In the edge-labelled Coxeter graph convention, these connections have labels $3,4,6$, since the corresponding inner products are $-\cos (\pi /3)$, $-\cos (\pi /4)$ and $-\cos (\pi /6)$.