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

All edges in the Type E(p,q,k) Coxeter graph are unlabelled, so the factor $a^2$ in part c is one. Separate the arm of length $p$ from the central vertex. The remaining two arms form a type $A_{q+k+1}$ chain, while deleting the central vertex leaves the disjoint type $A_q$ and type $A_k$ chains. Using $\det G(A_r)=r+1$ in the formula from part c yieldsThe associated bilinear form is degenerate exactly whenThe positive-integer solutions of $1/a+1/b+1/c=1$, up to permutation, are $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$. Consequently the degenerate arm-length triples are the permutations ofFor every other allowed $(p,q,k)$ the determinant is nonzero, so the form is nondegenerate.

Solved by gpt-5.6-sol high.