---- > [!definition] Definition. ([[Weyl chamber]]) > Let $(\Phi, E)$ be a [[root system]]. > If $\alpha \in \Phi$, let $H_{\alpha}=\{ x \in E: (x, \alpha) = 0 \}=\text{span}^{\perp}(\alpha)$. By Euclidean geometry, $E \setminus \bigcup_{\alpha \in \Phi} H_{\alpha} \neq \emptyset$. The [[connected component|connected components]] of this set are called **Weyl chambers**. > [[there exists exactly one root basis per Weyl chamber|Weyl chambers and root bases have the same data]]. ^definition > [!basicexample] > ![[Pasted image 20250417154955.png]] ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```