The root system and the Weyl group

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

$P= { λ∈𝔥ℝ*∣ ⟨λ,α∨⟩∈ℤ for allα∈ℝ } ,whereα∨= 2α⟨α,α⟩, (1.1)$

and the Weyl group $W=⟨{s}_{\alpha }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\alpha \in R⟩$ generated by reflections

$sα: 𝔥ℝ* ⟶ 𝔥ℝ* λ ⟼ λ-⟨λ,α∨⟩α (1.2)$

in the hyperplanes

$Hα= { x∈𝔥ℝ*∣ ⟨x,α∨⟩=0 } ,α∈R. (1.3)$

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 ${𝔥}_{ℝ}^{*}=ℝ\text{-span}\phantom{\rule{0.2em}{0ex}}\left(R\right),$
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 $R,$

$R+= { α∈R∣ ⟨x,α∨⟩>0 for allx∈C }$

and

$C= { x∈𝔥ℝ*∣ ⟨x,α∨⟩>0 for allα∈R+ } .$

For each $\alpha \in {R}^{+}$ define the raising operator ${R}_{\alpha }:\phantom{\rule{0.2em}{0ex}}P\to P$ by ${R}_{\alpha }\mu =\mu +\alpha \text{.}$ The dominance order on $P$ is given by

$μ≤λifλ= Rβ1… Rβℓμ (1.4)$

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

$R(w) = { α∈R+∣ Hαis betweenC andw-1C } ,and ℓ(w) = Card(R(w)) (1.5)$

is the length of $w.$ If ${R}^{-}=-{R}^{+}=\left\{-\alpha \phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\alpha \in {R}^{+}\right\}$ then

$R=R+∪R-and R(w)= { α∈R+∣ wα∈R- } ,forw∈W.$

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

$P = { λ∈𝔥ℝ*∣ ⟨λ,α∨⟩∈ ℤ for allα∈ R } , P+=P∩C‾ = { λ∈𝔥ℝ*∣ ⟨λ,α∨⟩∈ ℤ≥0 for allα∈ R+ } , P++=P∩C = { λ∈𝔥ℝ*∣ ⟨λ,α∨⟩∈ ℤ>0 for allα∈ R+ } , (1.6)$

where $\stackrel{‾}{C}=\left\{x\in {𝔥}_{ℝ}^{*}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}⟨x,{\alpha }^{\vee }⟩\ge 0\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}\alpha \in {R}^{+}\right\}$ 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

$P= ∑i=1n ℤωi, P+= ∑i=1n ℤ≥0ωi, andP++= ∑i=1n ℤ>0ωi. (1.7)$

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

$ρ= ∑i=1nωi= 12∑α∈R+ α,and the map P+⟶P++ λ⟼λ+ρ (1.8)$

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

$si2 = 1 for1≤i≤n, sisjsi… ⏟ mijfactors = sjsisj… ⏟ mijfactors , i≠j, (1.9)$

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

$R(w)= { αip,sip, αip-1,…, sip…si2 αi1 } .$

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).