Weyl groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 10 November 2012

Reflection presentation

A Weyl group is a finite $\mathbb{Z}$-reflection group $(W_0,{𝔥}_{\mathbb{Z}})$.

A finite $\mathbb{Z}-$reflection group is a pair $(W_0,{𝔥}_{\mathbb{Z}})$ where

1. ${𝔥}_{\mathbb{Z}}$ is a free $\mathbb{Z}$-module (has a $\mathbb{Z}$-basis $\{{\omega }_1,\dots ,{\omega }_l\}$)
2. $W_0$ is a finite subgroup of $GL\left({𝔥}_{\mathbb{Z}}\right)$ 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 $$\left(\begin{array}{cccc}\xi & & & \\ & 1& & 0\\ & & \ddots & \\ & 0& & 1\end{array}\right)with\xi \ne 1.$$ 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 The Weyl group, character lattice and cocharacter lattice of the pair $(G,T)$ are $$W_0=N/T,{𝔥}_{\mathbb{Z}}^{*}=\mathrm{Hom}(T,{ℂ}^{\times })\text{and}{𝔥}_{\mathbb{Z}}=\mathrm{Hom}({ℂ}^{\times },T),$$ respectively, where $\mathrm{Hom}(H,K)$ 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 $$N=\{g\in G|gT{g}^{-1}=T\}\text{is the normalizer of}T\text{in}G.$$ Since $W_0$ acts on $T$ (by conjugation) the group $W_0$ acts on ${𝔥}_{\mathbb{Z}}^{*}$ and ${𝔥}_{\mathbb{Z}}$ by $$\left(X^{w\lambda }\right)\left(t\right)=\left(w{X}^{\lambda }\right)\left(t\right)=X^{\lambda }\left(wt{w}^{-1}\right)\text{and}\left(Y^{w{\mu }^{\vee }}\right)\left(z\right)=\left(w{Y}^{{\mu }^{\vee }}\right)\left(z\right)=w\left(Y^{{\mu }^{\vee }}\right(z\left)\right)w^{-1},$$ for $w\in W_0$, $X^{\lambda }\in {𝔥}_{\mathbb{Z}}^{*}$, $Y^{{\mu }^{\vee }}\in {𝔥}_{\mathbb{Z}}$, $t\in T$, and $z\in {ℂ}^{\times }$.

Coxeter presentation

Let

1. $C_{\infty }$ be a fundamental region for the action of $W_0$ on ${𝔥}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}$,
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 $$s_i^2=1and{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{s_i{s}_j{s}_i\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{s_j{s}_i{s}_j\cdots }}}\text{for}i\ne j,$$ 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 $$\text{vertices}{\alpha }_1,\dots ,{\alpha }_n\text{and}\text{labeled edges}{\alpha }_i\stackrel{m_{ij}}{—}{\alpha }_j,$$ (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 $\mathbb{Z}$-reflection group is based on ??? in Andersen-Grodal etc al [AG+].

References

References?

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