# 4 – Aut | Chamber structure

$$\rl = \{ r \in \LL \mid \langle r,r \rangle_\LL = -2\}.$$ The collection $$\mathcal{C}_{\LL}=\left\{ \overline{C}\mid C\,\,\text{co. comp. of}\,\,~\ppp_{\LL}\,\,\backslash\bigcup_{v\in\mathcal{F}}(v)^{\perp},\,\,\text{Int}(\overline{C})\neq\emptyset\right\}$$ is called a chamber structure on $\mathcal{P}_{\LL}$.

A chamber is uniquely characterized by its Weyl vector