---- > [!definition] Definition. ([[Weyl group of a root system]]) > Let $(\Phi,E)$ be a crystallographic [[root system]]. The [[subgroup]] of the [[general linear group]] $\text{GL}(E)$ generated by the [[reflection|reflections]] $\{ w_{\alpha}:\alpha \in \Phi \}$ is called the **Weyl group of $\Phi$** and denoted as $W(\Phi)$ or just $W$. > > If $(\Phi, E)$ is not crystallographic then $W(\Phi)$ is still well-defined; in this case it is merely called the **reflection group of $\Phi$**. ^definition > [!basicproperties] > - $W(\Phi)$ is finite. ^properties **Finite.** Since $\Phi$ is stable under [[reflection|reflections]], $W$ [[group action|acts]] on $\Phi$ via $(w, \lambda)\mapsto w(\lambda)$. If $w \in W$ acts trivially on $\Phi$ (i.e., $w(\lambda)=\lambda$ for all $\lambda \in \Phi$), then we must have $w=\id_{E}$ because $\Phi$ spans $E$: $w(x)=x \ \fa x \in E$. Thus, the action of $W$ on $\Phi$ is [[faithful group action|faithful]], producing an [[injection]] $W \hookrightarrow \text{Perm }\Phi$. The latter set is finite, and thus the former is too. ---- #### ---- #### 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 > ```