= Solution
A <Weyl chamber> is a connected component of
$$
E\setminus\bigcup_{\alpha\in\Phi}\alpha^\perp.
$$
A <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 $(\gamma,\alpha)\ne0$ for every root. Declare
$$
\Phi_\gamma^+=\{\alpha\in\Phi:(\gamma,\alpha)>0\}.
$$
The 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.
Back to article page
