---- > [!definition] Definition. ([[root basis]]) > Let $(\Phi,E)$ be a [[root system]]. We call $\Delta \subset \Phi$ a **root basis** if > **1.** $\Delta$ is an $\mathbb{R}$-[[basis]] of $E$; > **2.** Writing $\Delta=\{ \alpha_{1}, \dots , \alpha_{\ell} \}$, for every $\alpha \in \Phi$ we can write $\alpha=\sum_{i=1}^{\ell}c_{i} \alpha_{i} \text{ where either } \{ c_{i} \} \subset \mathbb{Z}_{\geq 0} \text{ or } \{ c_{i} \}\subset \mathbb{Z}_{\leq 0}.$ Elements of $\Delta$ are called **simple roots**. Elements of $\Phi$ for which $\{ c_{i} \} \subset \mathbb{Z}_{\geq 0}$ called **positive roots**; the subset of positive roots is denoted $\Phi^{+}\subset \Phi$. The **root height** of $\alpha=\sum_{i}c_{i}\alpha_{i}$ is $\sum_{i}c_{i} \in \mathbb{Z}$; it can be positive or negative. A **highest root** is one with maximal root height among elements of $\Phi$. > [[There exists exactly one root basis per Weyl chamber]]. ^definition ---- #### ---- #### 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 > ```