Editing Root system (section)

==Weyl chambers and the Weyl group==
{{See also|Coxeter group#Affine Coxeter groups}}
[[File:Weyl_chambers_for_A2.png|class=skin-invert-image|thumb|right|The shaded region is the fundamental Weyl chamber for the base <math>\{\alpha_1,\alpha_2\}</math>]]
If <math>\Phi\subset E</math> is a root system, we may consider the hyperplane perpendicular to each root <math>\alpha</math>. Recall that <math>\sigma_\alpha</math> denotes the reflection about the hyperplane and that the [[Weyl group]] is the group of transformations of <math>E</math> generated by all the <math>\sigma_\alpha</math>'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a '''Weyl chamber'''. If we have fixed a particular set Δ of simple roots, we may define the '''fundamental Weyl chamber''' associated to Δ as the set of points <math>v\in E</math> such that <math>(\alpha,v)>0</math> for all <math>\alpha\in\Delta</math>.

Since the reflections <math>\sigma_\alpha,\,\alpha\in\Phi</math> preserve <math>\Phi</math>, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.

The figure illustrates the case of the <math>A_2</math> root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.

A basic general theorem about Weyl chambers is this:<ref>{{harvnb|Hall|2015|loc=Propositions 8.23 and 8.27}}</ref>
:'''Theorem''': The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.
In the <math>A_2</math> case, for example, the Weyl group has six elements and there are six Weyl chambers.

A related result is this one:<ref>{{harvnb|Hall|2015|loc=Proposition 8.29}}</ref>
:'''Theorem''': Fix a Weyl chamber <math>C</math>. Then for all <math>v\in E</math>, the Weyl-orbit of <math>v</math> contains exactly one point in the closure <math>\bar C</math> of <math>C</math>.