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

Let $\varphi =2\cos (\pi /5)=(1+\sqrt{5})/2$. In the vertex order along the displayed $3$-$5$-$3$ path, the Coxeter Gram matrix has diagonal entries $2$ and successive off-diagonal entries $-1,-\varphi ,-1$. Its leading principal determinants satisfyThe last determinant is nonzero, so the form is nondegenerate. Its negative determinant rules out positive semidefiniteness and hence also positive definiteness. Thus the answers are respectively no, no, and yes.

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The displayed simply-laced tree has arms of lengths $2$, $1$, and $3$ from its trivalent vertex, so it is the finite type $E_7$ Coxeter graph. Its Gram matrix is the $E_7$ Cartan matrix, which has positive leading principal minors in a leaf-removal ordering and determinant $2$. By Sylvester's criterion it is positive definite. It is therefore positive semidefinite and nondegenerate as well: the three answers are yes, yes, and yes.

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For the four-cycle, the quadratic form isIt is nonnegative, but it vanishes on the nonzero vector $(1,1,1,1)$. Equivalently, the Gram eigenvalues are $0,2,2,4$. The form is positive semidefinite, not positive definite, and degenerate: the three answers are no, yes, and no. This is the affine Coxeter graph ${\tilde{A}}_3$.

Solved by gpt-5.6-sol high.