## The affine Weyl group

Last update: 26 June 2012

## The affine Weyl group

This section is a summary of the main facts and notations that are needed for working with the affine Weyl group $\stackrel{˜}{W}.$ The main point is that the elements of the affine Weyl group can be identified with alcoves via the bijections in (2.11).

Let ${𝔥}_{ℝ}^{*}$ be a finite dimensional vector space over $ℝ.$ A reflection is a diagonalizable element of $\mathrm{GL}\left({𝔥}_{ℝ}^{*}\right)$ which has exactly one eigenvalue not equal to 1. A lattice is a free $ℤ-$module. A Weyl group is a finite subgroup $W$ of $\mathrm{GL}\left({𝔥}_{ℝ}^{*}\right)$ which is such that ${𝔥}_{ℝ}^{*}=L{\otimes }_{ℤ}ℝ.$ Let ${R}^{+}$ be an index set for the reflections in $W$ so that, for $\alpha \in {R}^{+},$ $sα is the reflection in the hyperplane Hα = (𝔥ℝ*)sα,$ the fixed point space of the transformation ${s}_{\alpha }.$ The chambers are the connected components of the complement $𝔥ℝ* ∖ ⋃α∈R+ Hα$ of these hyperplanes in ${𝔥}_{ℝ}^{*}.$ These are fundamental regions for the action of $W.$

Let $⟨\phantom{A},\phantom{A}⟩$ be a nondegenerate $W-$invariant bilinear form on ${𝔥}_{ℝ}^{*}.$ Fix a chamber $C$ and choose vectors ${\alpha }^{\vee }\in {𝔥}_{ℝ}^{*}$ such that $C = { x∈𝔥ℝ* | ⟨x,α∨⟩>0 } and P⊇L⊇Q, (AffW 1)$ where $P = { λ∈𝔥ℝ* | ⟨λ,α∨⟩∈ℤ } and Q = ∑α∈R+ℤα, where α = 2α∨ ⟨α∨,α∨⟩ . (AffW 2)$

Pictorially,

The alcoves are the connected components of the complement $𝔥ℝ* ∖ ⋃ α∈R+ j∈ℤ Hα,j of the (affine) hyperplanes Hα,j = { x∈𝔥ℝ* | ⟨x,α∨⟩=j }$ in ${𝔥}_{ℝ}^{*}.$ The fundamental alcove is the alcove $A⊆C such that 0∈A_, (AffW 3)$ where $\stackrel{_}{A}$ is the closure of $A.$ An example is the case of type ${C}_{2},$ where the picture is

The translation in $\lambda$ is the operator ${t}_{\lambda }:{𝔥}_{ℝ}^{*}\to {𝔥}_{ℝ}^{*}$ given by The reflection ${s}_{\alpha ,k}$ in the hyperplane ${H}_{\alpha ,k}$ is given by $sα,k = tkαsα = sαt-kα. (AffW 5)$ The extended affine Weyl group is $W˜ = P⋊W = { tλw | λ∈P, w∈W } with wtλ = twλw. (AffW 6)$ Denote the walls of $C$ by ${H}_{{\alpha }_{1}},...,{H}_{{\alpha }_{n}}$ and extend this indexing so that $Hα0 ,..., Hαn are the walls of A,$ the fundamental alcove. Then the affine Weyl group, $Waff = Q⋊W is generated by s0 ,..., sn, (AffW 7)$ the reflections in the hyperplanes ${H}_{{\alpha }_{0}},...,{H}_{{\alpha }_{n}}.$ Furthermore, $A$ is a fundamental region for te action of ${W}_{\mathrm{aff}}$ on ${𝔥}_{ℝ}^{*}$ and so there is a bijection $Waff → { alcoves in 𝔥ℝ* } w ↦ w-1A.$ The length of $w\in \stackrel{˜}{W}$ is $l(w) = number of hyperplanes between A and wA. (AffW 8)$ The difference between ${W}_{\mathrm{aff}}$ and $\stackrel{˜}{W}$ is the group $Ω = W˜/Waff ≅ P/Q. (AffW 9)$ The group $\Omega$ is the set of elements of $\stackrel{˜}{W}$ of length 0. An element of $\Omega$ acts on the fundamental alcove $A$ by an automorphism. Its action on $A$ induces a permutation of the walls of $A,$ and hence a permutation of $0,1,...,n.$ If $g\in \Omega$ and $g\ne 1$ let ${\omega }_{i}$ be the image of the origin under the action of $g$ on $A.$ If ${s}_{j}$ denotes the reflection in the ${j}^{\mathrm{th}}$ wall of $A$ and ${w}_{i}$ denotes the longest element of the stabilizer ${W}_{{\omega }_{i}}$ of ${\omega }_{i}$ in $W,$ then $gsig-1 = sg(i) and gw0wi = tωi. (AffW 10)$

The group $\stackrel{˜}{W}$ acts freely on $\Omega ×{𝔥}_{ℝ}^{*}$ ($|\Omega |$ copies of ${ℝ}^{n}$ tiled by alcoves) so that ${g}^{-1}A$ is in the same spot as $A$ except on the ${g}^{\mathrm{th}}$ "sheet" of $\Omega ×{𝔥}_{ℝ}^{*}.$ It is helpful to think of the elements of $\Omega$ as the deck transformations which transfer between the sheets in $\Omega ×{𝔥}_{ℝ}^{*}.$ Then $W˜ → { alcoves in Ω×𝔥ℝ* } w ↦ w-1A (AffW 11)$ is a bijection. In type ${C}_{2},$ the two sheets in $\Omega ×{𝔥}_{ℝ}^{*}$ look like $Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 (AffW 12a)$ and $Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 (AffW 12b)$ where the numbering on the walls of the alcoves is $\stackrel{˜}{W}$ equivariant so that, for $w\in \stackrel{˜}{W},$ the numbering on the walls of $wA$ is the $w$ image of the numbering on the walls of $A.$

The 0-polygon is the $W-$orbit of $A$ in $\Omega ×{𝔥}_{ℝ}^{*}$ and for $\lambda \in P,$

the $\lambda -$polygon is $\lambda +WA,$ $\phantom{\rule{4em}{0ex}}\begin{array}{c}\lambda \lambda +{s}_{1}A\lambda +A\lambda +{s}_{1}{s}_{2}A\lambda +{s}_{2}A\lambda +{s}_{1}{s}_{2}{s}_{1}A\lambda +{s}_{2}{s}_{1}A\lambda +{w}_{0}A\lambda +{s}_{2}{s}_{1}{s}_{2}A\end{array}$
the translate of the $W$ orbit of $A$ by $\lambda .$ The space $\Omega ×{𝔥}_{ℝ}^{*}$ is tiled by the polygons and, via (2.11 [Ref?]), we make identifications between $W,\stackrel{˜}{W},P$ and their geometric counterparts in $\Omega ×{𝔥}_{ℝ}^{*}:$ $W˜ = {alcoves}, W = { alcoves in the 0-polygon }, P = { centers of polygons }. (AffW 13)$

Define $P+ = P∩C_ and P++ = P∩C (AffW 14)$ so that ${P}^{+}$ is a set of representatives of the orbits of the action of $W$ on $P.$ The fundamental weights are the generators ${\omega }_{1},...,{\omega }_{n}$ of the ${ℤ}_{\ge 0}-$module ${P}^{+}$ so that $C = ∑i=1n ℝ≥0 ωi, P+ = ∑i=1n ℤ≥0 ωi, and P++ = ∑i=1n ℤ>0 ωi. (AffW 15)$ The lattice $P$ has $ℤ-$basis ${\omega }_{1},...,{\omega }_{n}$ and the map $P+ → P++ λ ↦ ρ+λ, where ρ = ω1 +⋯+ ωn, (AffW 16)$ is a bijection. The simple coroots are ${\alpha }_{1}^{\vee },...,{\alpha }_{n}^{\vee }$ the dual basis to the fundamental weights, $⟨ ωi, αj∨ ⟩ = δij. (AffW 17)$ Define $C∨_ = ∑i=1n ℝ≤0 αi∨ and C∨ = ∑i=1n ℝ<0 αi∨. (AffW 18)$ The dominance order is the partial order on ${𝔥}_{ℝ}^{*}$ given by $μ≤λ if μ∈λ+C∨_. (AffW 19)$

In type ${C}_{2}$ the lattice $P=ℤ{\epsilon }_{1}+ℤ{\epsilon }_{2}$ with $\left\{{\epsilon }_{1},{\epsilon }_{2}\right\}$ an orthonormal basis of ${𝔥}_{ℝ}^{*}\cong {ℝ}^{2}$ and $W=\left\{1,{s}_{1},{s}_{2},{s}_{1}{s}_{2},{s}_{2}{s}_{1},{s}_{1}{s}_{2}{s}_{1},{s}_{2}{s}_{1}{s}_{2},{s}_{1}{s}_{2}{s}_{1}{s}_{2}\right\}$ is the dihedral group of order 8 generated by the reflections ${s}_{1}$ and ${s}_{2}$ in the hyperplanes ${H}_{{\alpha }_{1}}$ and ${H}_{{\alpha }_{2}},$ respectively, where $Hα1 = { x∈𝔥ℝ* | ⟨x,ε1⟩=0 } and Hα2 = { x∈𝔥ℝ* | ⟨x, ε2-ε1⟩ = 0 }.$

$Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 = ω2 ω1 ε1 0 Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 ε1 ρ The set P+ The set P++$

In this case $ω1 = ε1+ε2, α1 = 2ε1, α1∨ = ε1, ω2 = ε2, α2 = ε2-ε1, α2∨ = α2,$ and $R = { ±α1, ±α2, ±(α1+α2), ±(α1+2α2) }.$

## Notes and References

The above notes are taken from section 2 of the paper

[Ram] A. Ram, Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (2006), 963-1013.

They are also a rtyping into MathML of the notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notes2005/affWeyl12.14.05.pdf

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