Positive systems, bases, and chambers

Each chamber $C$ determines a positive system: by [L1] the sign of $(x,\alpha )$ is constant on $C$, so the set ${\mathrm{\Phi }}^{+}(C)=\{\alpha :(x,\alpha )>0\text{ for }x\in C\}$ is well defined independently of the chosen $x\in C$; it contains exactly one of $\pm \alpha$, hence arises from any $x\in C$ viewed as a regular vector via the inner product, and is a positive system in the sense of [L2].

Conversely each positive system ${\mathrm{\Phi }}^{+}$ determines a chamber $C({\mathrm{\Phi }}^{+})=\{x:(x,\alpha )>0\text{ for all }\alpha \in {\mathrm{\Phi }}^{+}\}$: it is nonempty because it contains the regular vector defining ${\mathrm{\Phi }}^{+}$; it is an open convex cone defined by finitely many strict linear inequalities, hence is contained in a single chamber; and it equals that chamber because no root changes sign strictly inside it and every boundary point lies in some root hyperplane.

Positive systems correspond bijectively to their simple-root bases: a base $\mathrm{\Delta }$ determines the positive system $\{\beta \in \mathrm{\Phi }:\beta ={\sum }_{\alpha \in \mathrm{\Delta }}{n}_{\alpha }\alpha \text{ with all }{n}_{\alpha }\in {\mathbb{Z}}_{\ge 0}\}$, and the positive system determines the base as its indecomposable elements, by [L2]; these are inverse constructions.

The two assignments are inverse: ${\mathrm{\Phi }}^{+}(C({\mathrm{\Phi }}^{+}))={\mathrm{\Phi }}^{+}$ because the roots positive on $C({\mathrm{\Phi }}^{+})$ are exactly those in ${\mathrm{\Phi }}^{+}$, one sign being constant on the cone; and $C({\mathrm{\Phi }}^{+}(C))=C$ because both are open convex sets defined by the same sign conditions and the sign pattern determines the chamber.

The Weyl group acts on chambers, hence by steps 1.1 and 2.1 on positive systems and bases, and the action is simply transitive by [L3]. Explicitly, $w\cdot {\mathrm{\Phi }}^{+}(C)={\mathrm{\Phi }}^{+}(wC)$ and a chamber has a unique Weyl image, so each positive system and base has exactly one Weyl translate, giving simple transitivity on all three sets.