---- > [!definition] Definition. ([[dominant weight in a root system]]) > Let $(\Phi, E)$ be a [[root system]], $\Delta$ a [[root basis]], and $X$ the [[root and weight lattice of a root system|weight lattice]] of $\Phi$. Let $\{\omega_{i}\}$ be the unique elements (the [[root and weight lattice of a root system|fundamental weights]]) of $E$ satisfying $\langle \omega_{i}, \check \alpha_{j} \rangle=\delta_{ij}$ for all $\alpha_{j} \in \Delta$. > An element $\lambda \in X$ is called a **dominant weight** [[reflection|if]] $\langle \lambda, \check\alpha \rangle \geq 0$ for all $\alpha \in \Phi$. > [!equivalence] > $\begin{align} \lambda \in X \text{ is dominant } &\iff \langle \lambda, \check \alpha_{i} \rangle \geq 0 \text{ for all } \alpha_{i} \in \Delta \\ & \iff \lambda = \sum_{i} c_{i} \omega_{i} \text{ with } c_{i} \in \mathbb{Z}_{\geq 0} \\ & \iff \lambda \text{ lies in the closure of the fundamental } \\ & \ \ \ \ \ \ \ \text{Weyl chamber corresponding to }\Delta. \end{align}$ > [!basicexample] The nonzero dominant weights for $A_{2}$ are pictured in dark blue below. ![[Pasted image 20250515171507.png]] ---- #### - [[there exists exactly one root basis per Weyl chamber|fundamental Weyl chamber]] ---- #### 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 > ```