Last update: 10 November 2012
A Weyl group is a finite -reflection group
A finite reflection group is
a pair where
- is a free
-module (has a -basis
- is a finite subgroup of generated by reflections.
is a matrix
is a reflection in a finite subgroup
has finite order and
is a root of unity.
The Weyl group, character lattice and
cocharacter lattice of the pair
where is the abelian group of algebraic group homomorphisms from with product given by pointwise multiplication,
(by conjugation) the group
be a fundamental region for the action of on
the walls of and
- the corresponding reflections.
is presented by generators
is the angle between
The Dynkin diagram, or Coxeter diagram,
of is the graph with
(the graph of the "1-skeleton of ").
Notes and References
These notes are intended to supplement various lecture series given by Arun Ram.
The definition of -reflection group is based on ???
in Andersen-Grodal etc al [AG+].