----- > [!proposition] Proposition. ([[the Weyl group and root bases]]) > Let $(\Phi, E)$ be a [[root system]], $W=W(\Phi)$ the corresponding [[Weyl group of a root system|Weyl group]]. > > 1. If $\Delta$ is a [[root basis]], then so is $w(\Delta)$ for all $w \in W$. Thus, $W$ [[group action|acts]] on the *set* of [[root basis|root bases]] of $\Phi$. *In fact, this action is [[simply transitive group action|simply transitive]].* > > 2. As a root-preserving [[linear isometry|isometry]], $W$ preserves the set of [[reflection|root hyperplanes]] $\{ H_{\alpha} \}_{\alpha \in \Phi}$. Thus, $W$ [[group action|acts]] on the set of [[Weyl chamber|Weyl chambers]] in $E$. > > 3. Moreover, the two actions are identified under the [[there exists exactly one root basis per Weyl chamber|bijection between]] [[root basis|root bases]] and [[Weyl chamber|Weyl chambers]]: $w(\mathscr{C}(\Delta))=\mathscr{C}(w(\Delta)),$ > where $\mathscr{C}(\Delta)$ denotes the [[there exists exactly one root basis per Weyl chamber|fundamental Weyl chamber]] attached to $\Delta$. > > 4. If $\Delta \subset \Phi$ is a fixed [[root basis]] and $\alpha \in \Phi$, then there exists $w \in W$ such that $w(\alpha) \in \Delta$: any root can by reflected into (possibly with many reflections) a fixed root basis. > 5. $W$ is [[subgroup generated by an element of a group|generated by]] *simple* reflections $\{ w_{\alpha_{i}}: \alpha_{i} \in \Delta \}$, where $\Delta$ is a fixed [[root basis]]. > [!proof]- Proof. ([[the Weyl group and root bases]]) > (1) minus the simply transitive part, (2), and (3) are fairy immeidate: > > **1, minus simply transitive part.** Certainly $w(\Delta)$ is a [[basis]] whose elements are roots, since $W$ is [[generating set of a group|generated by]] [[reflection|reflections]] and any [[reflection]] (a) preserves roots, by definition of [[root system]] & (b) belongs to $\text{GL}(E)$. And since it is linear, $w$ preserves nonnegative resp. nonpositive [[linear combination|linear combinations]]. > > **3.** $w \in W$ maps root hyperplanes to root hyperplanes because (a) it maps roots to roots by definition of root system (b) if $x \in E$ is orthogonal $\alpha \in \Phi$ then $w(x)$ is orthogonal to $w(\alpha)$, as $w$ is a [[linear isometry]]. > > - [ ] check compatibility (3) > > > > > - [ ] **4.** > > - [ ] **5.** > > - [ ] simply transitivitiy in **1** > I did these on iPad, bring over later as good review. > ----- #### ---- #### 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 > ```