The dual action, chambers, faces, and root hyperplanes

Let $S$, $m$, $V$, $B$, $W$, $\rho$, $\mathrm{\Phi }$ be as in The canonical reflection homomorphism, roots, reflections, and the positive cone, and let $V^{\ast }$ be the algebraic dual of $V$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$).

(1) The dual action. For $w\in W$ and $f\in V^{\ast }$ define $w\cdot f\in V^{\ast }$ by $(w\cdot f)(v):=f(\rho (w{)}^{-1}v)\mathrm{.}$ This is the dual (contragredient) action of $\rho$; that it really is a left action by linear maps is proved in The dual action, the faces, and the rank-two chamber tiling ↗. The dual action is used even when $B$ is degenerate: no identification of $V$ with $V^{\ast }$ through $B$ is made or assumed.

(2) Chambers, faces, root hyperplanes. The closed chamber is $C:=\{f\in V^{\ast }:f(e_s)\ge 0\text{ for all }s\in S\}$, its interior is $C^{\circ }:=\{f\in V^{\ast }:f(e_s)>0\text{ for all }s\in S\}$, and for $I\subseteq S$ the face $C_I$ is $C_I:=\{f\in V^{\ast }:f(e_s)=0\text{ for }s\in I\text{ and }f(e_s)>0\text{ for }s\mathrm{\notin }I\},$ the set of functionals of $C$ vanishing exactly on $I$. For a root $\alpha \in \mathrm{\Phi }$ the root hyperplane is $H_{\alpha }:=\{f\in V^{\ast }:f(\alpha )=0\}=\ker ({\mathrm{e}\mathrm{v}}_{\alpha })$, the kernel of the evaluation functional $f\mapsto f(\alpha )$ (Kernel and image of a linear map).

Neither the nonemptiness of the $C_I$ nor any orbit or tiling property is asserted here; existence of the faces and the exact rank-two chamber structure are proved in The dual action, the faces, and the rank-two chamber tiling ↗.