----- > [!proposition] Proposition. ([[there exists exactly one root basis per Weyl chamber]]) > Let $(\Phi, E)$ be a [[root system]]. For $\alpha \in \Phi$, let $H_{\alpha}=\{ x \in E: (x,\alpha)=0 \}=\text{span}^{\perp}(\alpha)$ be the $\alpha$-[[reflection|hyperplane]]. > > By Euclidean geometry, $E \setminus \bigcup_{\alpha \in \Phi}H_{\alpha}\neq \emptyset$. Take any $\gamma \in E \setminus \bigcup_{\alpha \in \Phi}H_{\alpha}$, and suggestively define $\begin{align} > \Phi_{\gamma}^{+}&:= \{ \alpha \in \Phi: (\alpha, \gamma) > 0 \} \\ > \Phi_{\gamma}^{-}&:= \{ \alpha \in \Phi: (\alpha, \gamma) < 0 \}. > \end{align}$ > Note that $\Phi=\Phi_{\gamma}^{+} \sqcup \Phi_{\gamma}^{-}$. (Draw a picture with $A_{2}$) > ![[Pasted image 20250510220159.png]] > > Say an element $\alpha \in \Phi_{\gamma}^{+}$ is **decomposable** if $\alpha=\beta_{1}+\beta_{2}$ for $\beta_{1},\beta_{2} \in \Phi_{\gamma}^{+}$ and **indecomposable** otherwise. Let $\Delta_{\gamma}$ be the set of indecomposable elements of $\Phi_{\gamma}^{+}$. Then: > > **1.** $\Delta_{\gamma}$ is a [[root basis]] with corresponding set of positive roots $\Phi_{\gamma}^{+}$. > > **2.** *Any* [[root basis]] of $E$ is of this form for some $\gamma$. > > **3.** If $\gamma,\gamma' \in E \setminus \bigcup_{\alpha \in \Phi}H_{\alpha}$, then $\Delta_{\gamma}=\Delta_{\gamma'}$ if and only if $\gamma,\gamma'$ live in the same [[Weyl chamber]]. Thus, $\gamma \mapsto \Delta_{\gamma}$ defines a [[bijection]] $\{ \text{Weyl chambers} \} \leftrightarrow \{ \text{root bases of } \Phi \}.$ > Its inverse is given by $\Delta_{} \mapsto \mathscr{C}(\Delta)=\{ x \in E: (x, \alpha)>0 \ \fa \alpha \in \Delta \}.$ The set $\mathscr{C}(\Delta)$ is called the **fundamental Weyl chamber attached to $\Delta$**. > [!proposition] Corollary. > We extract a useful result of the proof (Claim 2) of the theorem: if $\Delta$ is a root basis and $\alpha, \beta \in \Delta$ are distinct, then $(\alpha, \beta) \leq 0$, i.e., the angle between $\alpha$ and $\beta$ is obtuse. ^proposition > [!proof]- Proof. ([[there exists exactly one root basis per Weyl chamber]]) > ~ - [ ] (note that lemma 2 from lecture is corollary four in [[angles and lengths in a root system]]) ----- #### ---- #### 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 > ```