The root system and the Weyl group

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

Last update: 25 September 2012

The root system and the Weyl group

Let 𝔥* be a real vector space with a nondegenerate symmetric bilinear form ,. The basic data is a reduced irreducible root system R (defined below) in 𝔥*. Associated to R are the weight lattice

P= { λ𝔥* λ,α for allα } ,whereα= 2αα,α, (1.1)

and the Weyl group W=sααR generated by reflections

sα: 𝔥* 𝔥* λ λ-λ,αα (1.2)

in the hyperplanes

Hα= { x𝔥* x,α=0 } ,αR. (1.3)

With these definitions R is reduced irreducible root system if it is a subset of 𝔥* such that

  1. R is finite, 0R and 𝔥*=-span(R),
  2. W permutes the elements of R, that is, wαR for wW and αR,
  3. W is finite,
  4. RP,
  5. if αR then the only other multiple of α in R is -α,
  6. 𝔥* is an irreducible W-module.

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

R+= { αR x,α>0 for allxC }


C= { x𝔥* x,α>0 for allαR+ } .

For each αR+ define the raising operator Rα:PP by Rαμ=μ+α. The dominance order on P is given by

μλifλ= Rβ1 Rβμ (1.4)

for some sequence of positive roots β1,,β R+.

The various fundamental chambers for the action of W on 𝔥* are the w-1C, wW. The inversion set of an element wW is

R(w) = { αR+ Hαis betweenC andw-1C } ,and (w) = Card(R(w)) (1.5)

is the length of w. If R-=-R+= { -α αR+ } then

R=R+R-and R(w)= { αR+ wαR- } ,forwW.

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

P = { λ𝔥* λ,α for allα R } , P+=PC = { λ𝔥* λ,α 0 for allα R+ } , P++=PC = { λ𝔥* λ,α >0 for allα R+ } , (1.6)

where C= { x𝔥* x,α0 for allαR+ } is the closure of the fundamental chamber C.

The simple roots are the positive roots α1,,αn such that the hyperplanes Hαi, 1in, are the walls of C. The fundamental weights, ω1,,ωnP, are given by ωi,αj =δij, 1i,jn, and

P= i=1n ωi, P+= i=1n 0ωi, andP++= i=1n >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= 12αR+ α,and the map P+P++ λλ+ρ (1.8)

is a bijection (see Proposition 2.3).

The simple reflections are si=sαi, for 1in. The Weyl group W has a presentation by generators s1,,sn and the relations

si2 = 1 for1in, sisjsi mijfactors = sjsisj mijfactors , ij, (1.9)

where π/mij is the angle between the hyperplanes Hαi and Hαj. A reduced word for wW is an expression w=si1sip 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=si1sip is a reduced expression for w then p=(w).

([Bou1981, VI § no. 6 Cor. 2 to Prop. 17]) Let w=si1sip be a reduced word for w. Then

R(w)= { αip,sip, αip-1,, sipsi2 αi1 } .

The Bruhat order, or Bruhat-Chevalley order (see [Ste1968, § App., p. 126]), is the partial order on W such that vw if there is a reduced word for v, v=sj1sjk, which is a subword of a reduced word for w, w=si1sip, (that is, sj1sjk is a subsequence of the sequence si1sip).


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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