Weyl group#Weyl chambers

In mathematics, the Weyl chamber (named after Hermann Weyl ) a term from the theory of Lie groups. Weyl chambers are required for the positive and simple roots definition, they also play a central role in the theory of buildings.

Definition

Let be a finite-dimensional semisimple Lie algebra, a Cartan subalgebra and the corresponding root system.

For a root call

The corresponding hyperplane.

Then the connected components of hot

The Weyl chambers of the root system.

Action of the Weyl group

The Weyl group of works on and permutes the set of Weyl chambers, ie the action of the Weyl group on the set of Weyl chambers is simply transitive, and the number of Weyl chambers is the cardinality of the Weyl group.

The completion of a Weyl chamber is a fundamental domain for the action of the Weyl group.

Example

It should be

And

The corresponding root system consists of the six roots

According to

The ' s are three lines in two-dimensional vector space, decompose in six Weyl chambers.

The Weyl group in this case is the symmetric group, it permutes the six Weyl chambers.