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 is called dominant.

The interior of $𝒞$ is a fundamental domain for the action of the Weyl group on $E$. The image $w(𝒞)$ of $𝒞$ 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
Canonical name	WeylChamber
Date of creation	2013-03-22 13:12:00
Last modified on	2013-03-22 13:12:00
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	8
Author	rmilson (146)
Entry type	Definition
Classification	msc 17B20
Defines	positive Weyl chamber
Defines	dominant weight