The affine Weyl group

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

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 $\tilde{W}.$ 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 $\mathbb{Z}-$module. A Weyl group is a finite subgroup $W$ of $\mathrm{GL}\left({𝔥}_{ℝ}^{*}\right)$ which is $$generated\; by\; reflectionsandacts\; on\; a\; lattice\; L \; in\; {𝔥}_{ℝ}^{*}$$ such that ${𝔥}_{ℝ}^{*}=L{\otimes }_{\mathbb{Z}}ℝ.$ Let $R^{+}$ be an index set for the reflections in $W$ so that, for $\alpha \in R^{+},$ $$s_{\alpha }is\; the\; reflection\; in\; the\; hyperplane{H}_{\alpha }={\left({𝔥}_{ℝ}^{*}\right)}^{s_{\alpha }},$$ the fixed point space of the transformation $s_{\alpha }.$ The chambers are the connected components of the complement $${𝔥}_{ℝ}^{*}\setminus \left(\bigcup _{\alpha \in R^{+}}{H}_{\alpha }\right)$$ 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 $$\begin{array}{ll}C=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\alpha }^{\vee }⟩>0\}andP\supseteq L\supseteq Q,& (AffW\; 1)\end{array}$$ where $$\begin{array}{ll}P=\{\lambda \in {𝔥}_{ℝ}^{*}|⟨\lambda ,{\alpha }^{\vee }⟩\in \mathbb{Z}\}andQ=\sum _{\alpha \in R^{+}}\mathbb{Z}\alpha ,where\alpha =\frac{2{\alpha }^{\vee }}{⟨{\alpha }^{\vee },{\alpha }^{\vee }⟩}.& (AffW\; 2)\end{array}$$

Pictorially, $$⟨\lambda ,{\alpha }^{\vee }⟩is\; the\; distance\; from\; \lambda \; to\; the\; hyperplane\; H_{\alpha }.$$

The alcoves are the connected components of the complement $${𝔥}_{ℝ}^{*}\setminus \left(\bigcup _{\genfrac{}{}{0ex}{}{\alpha \in R^{+}}{j\in \mathbb{Z}}}{H}_{\alpha ,j}\right)of\; the\; (affine)\; hyperplanes{H}_{\alpha ,j}=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\alpha }^{\vee }⟩=j\}$$ in ${𝔥}_{ℝ}^{*}.$ The fundamental alcove is the alcove $$\begin{array}{ll}A\subseteq Csuch\; that0\in \stackrel{\_}{A},& (AffW\; 3)\end{array}$$ 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 $$\begin{array}{ll}{t}_{\lambda }\left(x\right)=\lambda +x,for\; \lambda \in P, x\in {𝔥}_{ℝ}^{*}.& (AffW\; 4)\end{array}$$ The reflection $s_{\alpha ,k}$ in the hyperplane $H_{\alpha ,k}$ is given by $$\begin{array}{ll}{s}_{\alpha ,k}=t_{k\alpha }{s}_{\alpha }=s_{\alpha }{t}_{-k\alpha }.& (AffW\; 5)\end{array}$$ The extended affine Weyl group is $$\begin{array}{ll}\tilde{W}=P⋊W=\left\{t_{\lambda }w\right|\lambda \in P,w\in W\}withw{t}_{\lambda }=t_{w\lambda }w.& (AffW\; 6)\end{array}$$ Denote the walls of $C$ by $H_{{\alpha }_1},...,H_{{\alpha }_n}$ and extend this indexing so that $$H_{{\alpha }_0},...,H_{{\alpha }_n}are\; the\; walls\; ofA,$$ the fundamental alcove. Then the affine Weyl group, $$\begin{array}{ll}{W}_{\mathrm{aff}}=Q⋊Wis\; generated\; by{s}_0,...,s_n,& (AffW\; 7)\end{array}$$ 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 $$\begin{array}{ccc}{W}_{\mathrm{aff}}& \to & \left\{alcoves\; in{𝔥}_{ℝ}^{*}\right\}\\ w& \mapsto & w^{-1}A.\end{array}$$ The length of $w\in \tilde{W}$ is $$\begin{array}{ll}l\left(w\right)=number\; of\; hyperplanes\; betweenAandwA.& (AffW\; 8)\end{array}$$ The difference between $W_{\mathrm{aff}}$ and $\tilde{W}$ is the group $$\begin{array}{ll}\Omega =\tilde{W}/W_{\mathrm{aff}}\cong P/Q.& (AffW\; 9)\end{array}$$ The group $\Omega$ is the set of elements of $\tilde{W}$ 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 $$\begin{array}{ll}g{s}_i{g}^{-1}=s_{g\left(i\right)}andg{w}_0{w}_i=t_{{\omega }_i}.& (AffW\; 10)\end{array}$$

The group $\tilde{W}$ acts freely on $\Omega \times {𝔥}_{ℝ}^{*}$ ($\left|\Omega \right|$ 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 $\Omega \times {𝔥}_{ℝ}^{*}.$ It is helpful to think of the elements of $\Omega$ as the deck transformations which transfer between the sheets in $\Omega \times {𝔥}_{ℝ}^{*}.$ Then $$\begin{array}{ll}\begin{array}{rcl}\tilde{W}& \to & \{alcoves\; in\Omega \times {𝔥}_{ℝ}^{*}\}\\ w& \mapsto & w^{-1}A\end{array}& (AffW\; 11)\end{array}$$ is a bijection. In type $C_2,$ the two sheets in $\Omega \times {𝔥}_{ℝ}^{*}$ look like $$\begin{array}{ll}\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_0} \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \]\end{array}& (AffW\; 12a)\end{array}$$ and $$\begin{array}{ll}\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_0} \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 1 \] \[ 0 \] \[ 1 \] \[ 0 \]\end{array}& (AffW\; 12b)\end{array}$$ where the numbering on the walls of the alcoves is $\tilde{W}$ equivariant so that, for $w\in \tilde{W},$ 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 $\Omega \times {𝔥}_{ℝ}^{*}$ and for $\lambda \in P,$

the translate of the $W$ orbit of $A$ by $\lambda .$ The space $\Omega \times {𝔥}_{ℝ}^{*}$ is tiled by the polygons and, via (2.11 [Ref?]), we make identifications between $W,\tilde{W},P$ and their geometric counterparts in $\Omega \times {𝔥}_{ℝ}^{*}:$ $$\begin{array}{ll}\tilde{W}=\left\{alcoves\right\},W=\left\{alcoves\; in\; the\; 0-polygon\right\},P=\left\{centers\; of\; polygons\right\}.& (AffW\; 13)\end{array}$$

Define $$\begin{array}{ll}{P}^{+}=P\cap \stackrel{\_}{C}and{P}^{++}=P\cap C& (AffW\; 14)\end{array}$$ 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 ${\mathbb{Z}}_{\ge 0}-$module $P^{+}$ so that $$\begin{array}{ll}C=\sum _{i=1}^n{ℝ}_{\ge 0}{\omega }_i,P^{+}=\sum _{i=1}^n{\mathbb{Z}}_{\ge 0}{\omega }_i,and{P}^{++}=\sum _{i=1}^n{\mathbb{Z}}_{>0}{\omega }_i.& (AffW\; 15)\end{array}$$ The lattice $P$ has $\mathbb{Z}-$basis ${\omega }_1,...,{\omega }_n$ and the map $$\begin{array}{ll}\begin{array}{ccc}{P}^{+}& \to & P^{++}\\ \lambda & \mapsto & \rho +\lambda ,\end{array}where\rho ={\omega }_1+\cdots +{\omega }_n,& (AffW\; 16)\end{array}$$ is a bijection. The simple coroots are ${\alpha }_1^{\vee },...,{\alpha }_n^{\vee }$ the dual basis to the fundamental weights, $$\begin{array}{ll}⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}.& (AffW\; 17)\end{array}$$ Define $$\begin{array}{ll}\stackrel{\_}{C^{\vee }}=\sum _{i=1}^n{ℝ}_{\le 0}{\alpha }_i^{\vee }and{C}^{\vee }=\sum _{i=1}^n{ℝ}_{<0}{\alpha }_i^{\vee }.& (AffW\; 18)\end{array}$$ The dominance order is the partial order on ${𝔥}_{ℝ}^{*}$ given by $$\begin{array}{ll}\mu \le \lambda if\mu \in \lambda +\stackrel{\_}{C^{\vee }}.& (AffW\; 19)\end{array}$$

In type $C_2$ the lattice $P=\mathbb{Z}{\epsilon }_1+\mathbb{Z}{\epsilon }_2$ with $\{{\epsilon }_1,{\epsilon }_2\}$ an orthonormal basis of ${𝔥}_{ℝ}^{*}\cong {ℝ}^2$ and $W=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,s_1{s}_2{s}_1,s_2{s}_1{s}_2,s_1{s}_2{s}_1{s}_2\}$ 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 $$H_{{\alpha }_1}=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\epsilon }_1⟩=0\}and{H}_{{\alpha }_2}=\{x\in {𝔥}_{ℝ}^{*}|⟨x,{\epsilon }_2-{\epsilon }_1⟩=0\}.$$

In this case $$\begin{array}{ccccccccc}{\omega }_1& =& {\epsilon }_1+{\epsilon }_2,& {\alpha }_1& =& 2{\epsilon }_1,& {\alpha }_1^{\vee }& =& {\epsilon }_1,\\ {\omega }_2& =& {\epsilon }_2,& {\alpha }_2& =& {\epsilon }_2-{\epsilon }_1,& {\alpha }_2^{\vee }& =& {\alpha }_2,\end{array}$$ and $$R=\{\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2),\pm ({\alpha }_1+2{\alpha }_2)\}.$$

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

References

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