## Weyl groups

Last update: 10 November 2012

## Reflection presentation

A Weyl group is a finite $ℤ$-reflection group $\left({W}_{0},{𝔥}_{ℤ}\right)$.

A finite $ℤ-$reflection group is a pair $\left({W}_{0},{𝔥}_{ℤ}\right)$ where

1. ${𝔥}_{ℤ}$ is a free $ℤ$-module (has a $ℤ$-basis $\left\{{\omega }_{1},\dots ,{\omega }_{l}\right\}$)
2. ${W}_{0}$ is a finite subgroup of $GL\left({𝔥}_{ℤ}\right)$ generated by reflections.
A reflection is a matrix $s\in {\mathrm{GL}}_{n}\left(ℂ\right)$ conjugate in ${\mathrm{GL}}_{n}\left(ℂ\right)$ to $( ξ 1 0 ⋱ 0 1 ) with ξ≠1.$ If $s$ is a reflection in a finite subgroup of ${\mathrm{GL}}_{n}\left(ℂ\right)$ then $s$ has finite order and $\xi$ 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 $\left(G,T\right)$ are $W0=N/T, 𝔥ℤ* = Hom(T,ℂ×) and 𝔥ℤ = Hom(ℂ×,T),$ respectively, where $\mathrm{Hom}\left(H,K\right)$ is the abelian group of algebraic group homomorphisms from $H\to K$ with product given by pointwise multiplication, $\left(\phi \psi \right)\left(h\right)=\phi \left(h\right)\psi \left(h\right)$, and $N={g∈G | gTg-1 =T} is the normalizer of T inG.$ Since ${W}_{0}$ acts on $T$ (by conjugation) the group ${W}_{0}$ acts on ${𝔥}_{ℤ}^{*}$ and ${𝔥}_{ℤ}$ by $(Xwλ) (t) = (wXλ) (t) = Xλ (wtw-1 ) and (Yw μ∨) (z) = (w Yμ∨ )(z) = w (Y μ∨ (z)) w-1 ,$ for $w\in {W}_{0}$, ${X}^{\lambda }\in {𝔥}_{ℤ}^{*}$, ${Y}^{{\mu }^{\vee }}\in {𝔥}_{ℤ}$, $t\in T$, and $z\in {ℂ}^{×}$.

## Coxeter presentation

Let

1. ${C}_{\infty }$ be a fundamental region for the action of ${W}_{0}$ on ${𝔥}_{ℝ}=ℝ{\otimes }_{ℤ}{𝔥}_{ℤ}$,
2. ${𝔥}^{{\alpha }_{1}},\dots ,{𝔥}^{{\alpha }_{n}}$ the walls of ${C}_{\infty }$ and
3. ${s}_{1},\dots ,{s}_{n}$ the corresponding reflections.

(Coxeter) ${W}_{0}$ is presented by generators ${s}_{1}...,{s}_{n}$ with relations $si2=1 and sisjsi⋯ ⏟ mij factors = sjsisj⋯ ⏟ mij factors for i≠j,$ where $\frac{\pi }{{m}_{ij}}={𝔥}^{{\alpha }_{i}}\measuredangle {𝔥}^{{\alpha }_{j}}$ is the angle between ${𝔥}^{{\alpha }_{i}}$ and ${𝔥}^{{\alpha }_{j}}$.

The Dynkin diagram, or Coxeter diagram, of ${W}_{0}$ is the graph with $vertices α1,…, αn and labeled edges αi —mij αj,$ (the graph of the "1-skeleton of ${C}_{\infty }$").

## Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. The definition of $ℤ$-reflection group is based on ??? in Andersen-Grodal etc al [AG+].

References?