Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 2 / c

For the same graph, let $e_0$ correspond to the central vertex and $e_1,\dots ,e_4$ to the leaves. The associated simply-laced Coxeter Gram matrix hasfor distinct leaves. The nonzero vectorsatisfies $(\delta ,e_i)=0$ for every basis vector. Thus the form is degenerate; in fact it is positive semidefinite with one-dimensional radical, as expected for the affine graph ${\tilde{D}}_4$.

The geometric form of a finite Coxeter group is positive definite. Since this form is degenerate, the group cannot be finite.

Solved by gpt-5.6-sol high.