Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 2 / a / Solution

A Weyl chamber is a connected component ofA root basis $\Delta$ is a basis of $E$ made of roots such that every root is an integer combination of $\Delta$ whose nonzero coefficients all have one sign.

Choose a regular vector $\gamma$, meaning $(\gamma ,\alpha )\ne 0$ for every root. DeclareThe indecomposable roots in ${\Phi }_{\gamma }^{+}$ form a root basis ${\Delta }_{\gamma }$, and every root basis arises in this way. Its chamber is the component containing $\gamma$.

Solved by gpt-5.6-sol high.