Weyl chamber

Let $E$ be a Euclidean vector space, $R\subset E$ a root system, and$R^{+}\subset R$ a choice of positive roots. We define the positive Weyl chamber (relative to $R^{+}$) to be the closed set

$$𝒞=\{u\in E\mid (u,\alpha )\ge 0\text{ for all }\alpha \in R^{+}\}.$$

A weight which lies inside the positive Weyl chamber iscalled dominant.

The interior of $𝒞$ is a fundamental domain for the actionof the Weyl group on $E$. The image $w(𝒞)$ of $𝒞$under the any element $w$ of the Weyl group is called a Weylchamber. The Weyl group $W$ acts simply transitively on the set ofWeyl chambers.