Generator of Reflection Group

Height of root relative to simple system

$\beta \in \Phi$, write uniquely $\beta ={\sum }_{\alpha \in \Delta }{c}_{\alpha }\alpha$, call $\sum c_{\alpha }$ the height of $\beta$(relative to $\Delta$), abbriviate $\text{ht}(\beta )$. $\text{ht}(\beta )=1$ for $\beta \in \Delta$.

Theorem

For a fixed simple system $\Delta$, $W$ is generated by the reflections $s_{\alpha }(\alpha \in \Delta )$.

Proof Denote $W^{\prime }$ by the subgroup of $W$ that generated by $s_{\alpha }(\alpha \in \Delta )$. We’ll show $W^{\prime }=W$. (1) For any $\beta \in \Pi$, there exists $\alpha \in \Delta$ that $\alpha \in W^{\prime }\beta$. If $\beta \in \Pi$, consider $W^{\prime }\beta \cap \Pi$, let $\gamma$ be the element in $W^{\prime }\beta \cap \Pi$ with smallest possible height, $\gamma ={\sum }_{\alpha \in \Delta }{c}_{\alpha }\alpha ,$ $0<(\gamma ,\gamma )={\sum }_{\alpha \in \Delta }{c}_{\alpha }(\gamma ,\alpha ),$ so there exist some $\alpha \in \Delta$ that $(\gamma ,\alpha )>0$. If $\gamma \ne \alpha$, consider $s_{\alpha }\gamma =\gamma -\frac{2(\gamma ,\alpha )}{(\alpha ,\alpha )}\alpha$, $s_{\alpha }\gamma \in W^{\prime }\beta \cap \Pi$ from proposition, and with $\text{ht}(s_{\alpha }\gamma )<\text{ht}(\gamma )$, we get a contradiction, so $\gamma =\alpha \in \Delta$. (2) $W^{\prime }\Delta =\Phi$ From (1), $\alpha =w\beta$ for some $w\in W^{\prime }$, then $\beta =w^{-1}\alpha$, for any positive $\beta$, so $\Pi \subset W^{\prime }\Delta$. When $\beta$ is negative, $-\beta$ is positive, $-\beta =w\alpha$ for some $w\in W^{\prime },\alpha \in \Delta$, $\beta =w{s}_{\alpha }\alpha$, $w{s}_{\alpha }\in W^{\prime }$, then $-\Pi \subset W^{\prime }\Delta$, together with $W^{\prime }\Delta \subset \Phi$, we get $W^{\prime }\Delta =\Phi$. (3) W’ = W Take $s_{\beta }$ as any generator of $W$, $\beta =w\alpha$ for $w\in W^{\prime }$, $\alpha \in \Delta$> $s_{\beta }=s_{w\alpha }=w{s}_{\alpha }{w}^{-1}\in W^{\prime }$, so $W=W^{\prime }$.

Corollary

Given $\Delta$, for every $\beta \in \Phi$ there exist $w\in W$, such that $w\beta \in \Delta$.

Relation of generators

Fix a simple system $\Delta$ in $\Phi$. Then $W$ is generated by the set $S:=\{s_{\alpha },\alpha \in \Delta \}$, subject only to the relations $(s_{\alpha }{s}_{\beta }{)}^{m(\alpha ,\beta )}=1(\alpha ,\beta \in \Delta ),$ with $m(\alpha ,\beta )$ denoting the order of $s_{\alpha }{s}_{\beta }$.