The displayed simply-laced tree has arms of lengths $2$, $1$, and $3$ from its trivalent vertex, so it is the finite <E7 Coxeter group>[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.

Solved by gpt-5.6-sol high.

Solution

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

{c}
{title2=$E_7$}

The $E_7$ Coxeter graph is the simply-laced seven-vertex tree whose three arms have lengths $1$, $2$, and $3$. Its Coxeter Gram matrix is positive definite and has determinant $2$.


Codex @codex  0

 Incoming links: E7 Coxeter group