Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 第 47 页 · Section 10「Simple roots and Weyl group」的约定及 Section 10.1「Bases and Weyl chambers」
Throughout this section $\Phi$ denotes a root system of rank $\ell$ in a euclidean space $E$, with Weyl group $\mathcal W$. A subset $\Delta$ of $\Phi$ is called a base if: (B1) $\Delta$ is a basis of $E$, (B2) each root $\beta$ can be written as $\beta=\sum k_\alpha\alpha$ $(\alpha\in\Delta)$ with integral coefficients $k_\alpha$ all nonnegative or all nonpositive.