Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / a / i / Solution

A Weyl chamber is a connected component ofA root basis $\Delta \subseteq \Phi$ is a vector-space basis of $E$ such that every root is an integer combination of elements of $\Delta$ with all nonzero coefficients of one sign.

To construct one, choose a regular vector $\gamma \in E$, meaning $(\gamma ,\alpha )\ne 0$ for every root. DeclareThe positive roots in ${\Phi }_{\gamma }^{+}$ which cannot be written as sums of two positive roots form a root basis ${\Delta }_{\gamma }$. Vectors $\gamma$ in the same Weyl chamber give the same basis.