# 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

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

A weight which lies inside the positive Weyl chamber is called dominant.

The interior of $\mathcal{C}$ is a fundamental domain for the action of the Weyl group on $E$. The image $w(\mathcal{C})$ of $\mathcal{C}$ under the any element $w$ of the Weyl group is called a Weyl chamber. The Weyl group $W$ acts simply transitively on the set of Weyl chambers.

Title Weyl chamber WeylChamber 2013-03-22 13:12:00 2013-03-22 13:12:00 rmilson (146) rmilson (146) 8 rmilson (146) Definition msc 17B20 positive Weyl chamber dominant weight