Weyl groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 10 November 2012

Reflection presentation

A Weyl group is a finite -reflection group (W0,𝔥).

A finite -reflection group is a pair (W0, 𝔥) where

  1. 𝔥 is a free -module (has a -basis {ω1,,ωl})
  2. W0 is a finite subgroup of GL(𝔥) generated by reflections.
A reflection is a matrix s GLn() conjugate in GLn() to ( ξ 1 0 0 1 ) with ξ1. If s is a reflection in a finite subgroup of GLn() then s has finite order and ξ is a root of unity.

N/T presentation

Let G a complex reductive algebraic group | T a maximal torus. The Weyl group, character lattice and cocharacter lattice of the pair (G,T) are W0=N/T, 𝔥* = Hom(T,×) and 𝔥 = Hom(×,T), respectively, where Hom(H,K) is the abelian group of algebraic group homomorphisms from HK with product given by pointwise multiplication, (φψ)(h) = φ(h)ψ(h) , and N={gG | gTg-1 =T} is the normalizer of T inG. Since W0 acts on T (by conjugation) the group W0 acts on 𝔥* and 𝔥 by (Xwλ) (t) = (wXλ) (t) = Xλ (wtw-1 ) and (Yw μ) (z) = (w Yμ )(z) = w (Y μ (z)) w-1 , for wW0, Xλ 𝔥*, Yμ 𝔥 , tT, and z×.

Coxeter presentation


  1. C be a fundamental region for the action of W0 on 𝔥= 𝔥,
  2. 𝔥α1 ,, 𝔥αn the walls of C and
  3. s1,, sn the corresponding reflections.

(Coxeter) W0 is presented by generators s1 ...,sn with relations si2=1 and sisjsi mij factors = sjsisj mij factors for ij, where π mij = 𝔥αi𝔥αj is the angle between 𝔥αi and 𝔥αj.

The Dynkin diagram, or Coxeter diagram, of W0 is the graph with vertices α1,, αn and labeled edges αi mij αj, (the graph of the "1-skeleton of C").

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+].



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