Open and closed Weyl chambers

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system (Reduced crystallographic Euclidean root system). For a root $\alpha$ the root hyperplane is $L_{\alpha }=\{x\in E:(x,\alpha )=0\}$. The complement $E\setminus {\bigcup }_{\alpha \in \mathrm{\Phi }}{L}_{\alpha }$ is a finite union of open convex cones, and its connected components are the open Weyl chambers of $\mathrm{\Phi }$. Each chamber is open and convex. A root hyperplane $L_{\alpha }$ is a wall of a chamber $D$ when $\bar{D}\cap L_{\alpha }$ contains a nonempty relatively open subset of $L_{\alpha }$—equivalently, when it is the supporting hyperplane of a codimension-one face of $\bar{D}$. In rank at least two, merely meeting the boundary at the common vertex $0$ does not make a root hyperplane a wall. Every chamber is the set of solutions of a system of strict homogeneous linear inequalities $\pm (x,\alpha )>0$.

Fix a positive system ${\mathrm{\Phi }}^{+}$ with simple roots $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots). The fundamental chamber is $C=\{x\in E:(x,{\alpha }_i)>0\text{ for }i=1,\dots ,r\},$ and its closure $\bar{C}$ is defined by the same inequalities with $\ge$ in place of $>$. The set $C$ is a chamber because it is a nonempty open convex cone on which no root vanishes: a positive root is a nonnegative integral combination of the ${\alpha }_i$ (Simple roots form a signed integral basis), so $(x,\alpha )>0$ for $x\in C$ and every positive root $\alpha$. The Weyl group $W(\mathrm{\Phi })$ (Weyl group) permutes the root hyperplanes and therefore permutes the open chambers; each $w\in W(\mathrm{\Phi })$ sends the closure of a chamber to the closure of its image chamber.