Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 1 / a / Solution

A root system in the real inner product space $V$ is a finite spanning set $\Phi \subset V\setminus \{0\}$ such thatwhere the orthogonal reflectionFor a crystallographic root system one additionally requires $2(\beta ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$; that condition is not needed for general finite reflection groups.

A fundamental system of a root system is a basis $\Delta \subset \Phi$ such that every root has either all nonnegative or all nonpositive coordinates in this basis. Its associated positive system of a root system isand $\Phi =\Pi \bigsqcup (-\Pi )$.

Solved by gpt-5.6-sol high.