The affine Weyl group

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

Last update: 25 September 2012

The affine Weyl group

For λP, the translation in λ is

tλ: 𝔥* 𝔥* x x+λ. (1.10)

The extended affine Weyl group W is the group

W= { wtλ|wW ,λP } , (1.11)

with multiplication determined by the relations

tλtμ= tλ+μ,and twλw=w tλ, (1.12)

for λ,μP and wW, and so W is a semidirect product of W and the group of translations {tλλP}. It is the group of transformations of 𝔥* generated by the sα, αR+, and tλ, λP. The affine Weyl group Waff is the subgroup of W generated by the reflections

sα,k: 𝔥*𝔥* in the hyperplanes Hα,k= { x𝔥* x,α=k } ,αR+,k. (1.13)

The reflections sα,k can be written as elements of W via the formula

sα,k= tkαsα= sαt-kα. (1.14)

The highest short root of R is the unique element φR+ such that the fundamental alcove

A=C { x𝔥* x,φ<1 } (1.15)

is a fundamental region for the action of Waff on 𝔥*. The various fundamental chambers for the action of Waff on 𝔥* are w-1A, wWaff. The inversion set of wW is

R(w)= { Hα,k Hα,kis betweenA andw-1 A } and (w)=Card (R(w))

is the length of w. If wW and λP then

(wtλ)= αR+ λ,α+ χ(wα) , (1.16)

where, for a root βR, set χ(β)=0, if βR+, and χ(β)=1, if βR-.

In type C2 we have the picture.


Hα0=Hφ,1 ands0= sφ,1=tϕ sϕ=sϕ t-ϕ, (1.17)

and let Hα1,,Hαn and s1,,sn be as in (1.9). Then the walls of A are the hyperplanes Hα0,Hα1,,Hαn and the group Waff has a presentation by generators s0,s1,,sn and relations

si2 = 1 for0in, sisjsi mijfactors = sjsisj mijfactors , ij, (1.18)

where π/mij is the angle between the hyperplanes Hαi and Hαj.

Let w0 be the longest element of W and let wi be the longest element of the subgroup Wωi= { wWwωi =ωi } . Let φ=c1α1+ +cnαn. Then let

Ω= { gW (g)=0 } ={1} {gici=1}, wheregi=tωi wiw0, (1.19)

(see [Bou1981, VI § no. 3 Prop. 6]). Each element gΩ sends the alcove A to itself and thus permutes the walls Hα0, Hα1,, Hαn of A. Denote the resulting permutation of {0,1,,n} also by g. Then

gsig-1= sg(i), for0in, (1.20)

and the group W is presented by the generators s0,s1,,sn and gΩ with the relations (1.18) and (1.20). In the setting of Example 1.1, Wω1={1,s2}, Wω2={1,s1}, w1=s2, w2=s1 and w0=s1s2s1s2, and φ=2α1+α2 so that c1=2, c2=1 and Ω={1,g2}/2, where g2=tω2s2s1s2.


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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