The root system and the Weyl group

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

Last update: 25 September 2012

The root system and the Weyl group

Let ${𝔥}_{ℝ}^{*}$ be a real vector space with a nondegenerate symmetric bilinear form $⟨,⟩\text{.}$ The basic data is a reduced irreducible root system $R$ (defined below) in ${𝔥}_{ℝ}^{*}\text{.}$ Associated to $R$ are the weight lattice

$$\begin{array}{cc}P=\{\lambda \in {𝔥}_{ℝ}^{*}\mid ⟨\lambda ,{\alpha }^{\vee }⟩\in \mathbb{Z}\text{for all}\alpha \in ℝ\},\text{where}{\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩},& \text{(1.1)}\end{array}$$

and the Weyl group $W=⟨s_{\alpha }\mid \alpha \in R⟩$ generated by reflections

$$\begin{array}{cc}\begin{array}{cccc}{s}_{\alpha }:& {𝔥}_{ℝ}^{*}& ⟶& {𝔥}_{ℝ}^{*}\\ & \lambda & ⟼& \lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha \end{array}& \text{(1.2)}\end{array}$$

in the hyperplanes

$$\begin{array}{cc}{H}_{\alpha }=\{x\in {𝔥}_{ℝ}^{*}\mid ⟨x,{\alpha }^{\vee }⟩=0\},\alpha \in R\text{.}& \text{(1.3)}\end{array}$$

With these definitions $R$ is reduced irreducible root system if it is a subset of ${𝔥}_{ℝ}^{*}$ such that

1. $R$ is finite, $0\notin R$ and ${𝔥}_{ℝ}^{*}=ℝ\text{-span}\left(R\right),$
2. $W$ permutes the elements of $R,$ that is, $w\alpha \in R$ for $w\in W$ and $\alpha \in R,$
3. $W$ is finite,
4. $R\subseteq P,$
5. if $\alpha \in R$ then the only other multiple of $\alpha$ in $R$ is $-\alpha ,$
6. ${𝔥}_{ℝ}^{*}$ is an irreducible $W$-module.

The choice of fundamental region $C$ for the action of $W$ on ${𝔥}_{ℝ}^{*}$ is equivalent to a choice of positive roots $R^{+}$ of $R,$

$$R^{+}=\{\alpha \in R\mid ⟨x,{\alpha }^{\vee }⟩>0\text{for all}x\in C\}$$

and

$$C=\{x\in {𝔥}_{ℝ}^{*}\mid ⟨x,{\alpha }^{\vee }⟩>0\text{for all}\alpha \in R^{+}\}\text{.}$$

For each $\alpha \in R^{+}$ define the raising operator $R_{\alpha }:P\to P$ by $R_{\alpha }\mu =\mu +\alpha \text{.}$ The dominance order on $P$ is given by

$$\begin{array}{cc}\mu \le \lambda \text{if}\lambda =R_{{\beta }_1}\dots R_{{\beta }_{\ell }}\mu & \text{(1.4)}\end{array}$$

for some sequence of positive roots ${\beta }_1,\dots ,{\beta }_{\ell }\in R^{+}\text{.}$

The various fundamental chambers for the action of $W$ on ${𝔥}_{ℝ}^{*}$ are the $w^{-1}C,$ $w\in W\text{.}$ The inversion set of an element $w\in W$ is

$$\begin{array}{cc}\begin{array}{ccc}R\left(w\right)& =& \{\alpha \in R^{+}\mid H_{\alpha }\text{is between}C\text{and}{w}^{-1}C\},\text{and}\\ \ell \left(w\right)& =& \text{Card}\left(R\left(w\right)\right)\end{array}& \text{(1.5)}\end{array}$$

is the length of $w.$ If $R^{-}=-R^{+}=\{-\alpha \mid \alpha \in R^{+}\}$ then

$$R=R^{+}\cup R^{-}\text{and}R\left(w\right)=\{\alpha \in R^{+}\mid w\alpha \in R^{-}\},\text{for}w\in W\text{.}$$

The weight lattice, the set of dominant integral weights, and the set of strictly dominant integral weights, are

$$\begin{array}{cc}\begin{array}{ccc}P& =& \{\lambda \in {𝔥}_{ℝ}^{*}\mid ⟨\lambda ,{\alpha }^{\vee }⟩\in \mathbb{Z}\text{for all}\alpha \in R\},\\ P^{+}=P\cap \bar{C}& =& \{\lambda \in {𝔥}_{ℝ}^{*}\mid ⟨\lambda ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}\text{for all}\alpha \in R^{+}\},\\ P^{++}=P\cap C& =& \{\lambda \in {𝔥}_{ℝ}^{*}\mid ⟨\lambda ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{>0}\text{for all}\alpha \in R^{+}\},\end{array}& \text{(1.6)}\end{array}$$

where $\bar{C}=\{x\in {𝔥}_{ℝ}^{*}\mid ⟨x,{\alpha }^{\vee }⟩\ge 0\text{for all}\alpha \in R^{+}\}$ is the closure of the fundamental chamber $C\text{.}$

The simple roots are the positive roots ${\alpha }_1,\dots ,{\alpha }_n$ such that the hyperplanes $H_{{\alpha }_i},$ $1\le i\le n,$ are the walls of $C\text{.}$ The fundamental weights, ${\omega }_1,\dots ,{\omega }_n\in P,$ are given by $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},$ $1\le i,j\le n,$ and

$$\begin{array}{cc}P=\sum _{i=1}^n\mathbb{Z}{\omega }_i,P^{+}=\sum _{i=1}^n{\mathbb{Z}}_{\ge 0}{\omega }_i,\text{and}{P}^{++}=\sum _{i=1}^n{\mathbb{Z}}_{>0}{\omega }_i\text{.}& \text{(1.7)}\end{array}$$

The set $P^{+}$ is an integral cone with vertex 0, the set $P^{++}$ is a integral cone with vertex

$$\begin{array}{cc}\rho =\sum _{i=1}^n{\omega }_i=\frac{1}{2}\sum _{\alpha \in R^{+}}\alpha ,\text{and the map}\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}& \text{(1.8)}\end{array}$$

is a bijection (see Proposition 2.3).

The simple reflections are $s_i=s_{{\alpha }_i},$ for $1\le i\le n\text{.}$ The Weyl group $W$ has a presentation by generators $s_1,\dots ,s_n$ and the relations

$$\begin{array}{cc}\begin{array}{cccc}{s}_i^2& =& 1& \text{for}1\le i\le n,\\ \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }& =& \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots },& i\ne j,\end{array}& \text{(1.9)}\end{array}$$

where $\pi /m_{ij}$ is the angle between the hyperplanes $H_{{\alpha }_i}$ and $H_{{\alpha }_j}\text{.}$ A reduced word for $w\in W$ is an expression $w=s_{i_1}\dots s_{i_p}$ for $w$ as a product of simple reflections which has $p$ minimal. The following lemma describes the inversion set in terms of the simple roots and the simple reflections and shows that if $w=s_{i_1}\dots s_{i_p}$ is a reduced expression for $w$ then $p=\ell \left(w\right)\text{.}$

([Bou1981, VI § no. 6 Cor. 2 to Prop. 17]) Let $w=s_{i_1}\dots s_{i_p}$ be a reduced word for $w\text{.}$ Then

$$R\left(w\right)=\{{\alpha }_{i_p},s_{i_p},{\alpha }_{i_{p-1}},\dots ,s_{i_p}\dots s_{i_2}{\alpha }_{i_1}\}\text{.}$$

The Bruhat order, or Bruhat-Chevalley order (see [Ste1968, § App., p. 126]), is the partial order on $W$ such that $v\le w$ if there is a reduced word for $v,$ $v=s_{j_1}\dots s_{j_k},$ which is a subword of a reduced word for $w,$ $w=s_{i_1}\dots s_{i_p},$ (that is, $s_{j_1}\dots s_{j_k}$ is a subsequence of the sequence $s_{i_1}\dots s_{i_p}\text{).}$

Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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