Affine Coxeter Group

Affine Group

We introduce the $\mathrm{Aff}(V)$, which is the semidirect product of $\mathrm{GL}(V)$ and the group of translations by elements of $V$. To each $\lambda \in V$ we associate the translation $t(\lambda )$, which sends $\mu \in V$ to $\mu +\lambda$. Then we see immediately that, for any $g\in \mathrm{GL}(V)$ and $\lambda \in V$, $gt(\lambda )g^{-1}=t(g\lambda ),$ showing that the group of translations is indeed normalized by $\mathrm{GL}(V)$.

Affine Hyperplane

For each root $\alpha$ and each integer $k$, define an affine hyperplane $H_{\alpha ,k}:=\{\lambda \in V\mid (\lambda ,\alpha )=k\}.$ Note that $H_{\alpha ,k}=H_{-\alpha ,-k}$ and that $H_{\alpha ,0}$ coincides with the reflecting hyperplane $H_{\alpha }$. Note too that $H_{\alpha ,k}$ can be obtained by translating $H_{\alpha }$ by $\frac{k}{2}{\alpha }^{\vee }$.

Affine Reflection

Define the corresponding affine reflection as follows: $s_{\alpha ,k}(\lambda ):=\lambda -((\lambda ,\alpha )-k){\alpha }^{\vee }.$

This is geometrically correct, because it fixes $H_{\alpha ,k}$ pointwise and sends the $0$ vector to $k{\alpha }^{\vee }$. We can also write $s_{\alpha ,k}$ as $t(k{\alpha }^{\vee })s_{\alpha }$. In particular, $s_{\alpha ,0}=s_{\alpha }$.

Denote by $\mathcal{H}$ the collection of all hyperplanes $H_{\alpha ,k}$ ($\alpha \in \Phi ,k\in \mathbf{Z}$). The following proposition (the proof of which is an immediate calculation) shows that the elements of $\mathcal{H}$ are permuted in a natural way by $W$ as well as by certain translations in $\mathrm{Aff}(V)$.

Affine Weyl Group

Let $\Phi$ be a crystallographic root system in a real vector space $V$, and let $W$ be its associated finite Weyl group. For each root $\alpha \in \Phi$ and integer $k\in \mathbb{Z}$, define the corresponding affine reflection $s_{\alpha ,k}:V\to V$ by: $s_{\alpha ,k}(\lambda ):=\lambda -((\lambda ,\alpha )-k){\alpha }^{\vee }$ The affine Weyl group, denoted by $W_a$, is defined as the subgroup of the affine transformations $\mathrm{Aff}(V)$ generated by the set of all such affine reflections: $W_a:=⟨s_{\alpha ,k}\mid \alpha \in \Phi ,k\in \mathbb{Z}⟩$

Proposition

• (a) If $w\in W$, then $w{H}_{\alpha ,k}=H_{w\alpha ,k}$ and $w{s}_{\alpha ,k}{w}^{-1}=s_{w\alpha ,k}$.
• (b) If $\lambda \in V$ satisfies $(\lambda ,\alpha )\in \mathbf{Z}$ for all roots $\alpha$, then $t(\lambda )H_{\alpha ,k}=H_{\alpha ,k+(\lambda ,\alpha )}$ and $t(\lambda )s_{\alpha ,k}t(-\lambda )=s_{\alpha ,k+(\lambda ,\alpha )}$.

Proposition

$W_a$ is the semidirect product of $W$ and the translation group corresponding to the coroot lattice $L=L({\Phi }^{\vee })$.

Extended Affine Weyl Group

Let $W$ be a finite Weyl group acting on a vector space $V$. Let $L$ denote the coroot lattice and $\hat{L}$ denote the coweight lattice (or weight lattice, depending on the dual setting), such that $L\subseteq \hat{L}$ and both lattices are $W$-stable. Because the finite Weyl group $W$ stabilizes the lattice $\hat{L}$, it normalizes the group of translations associated with $\hat{L}$. The extended affine Weyl group, denoted by ${\hat{W}}_a$, is defined as the semidirect product of the finite Weyl group $W$ and the translation group corresponding to $\hat{L}$: ${\hat{W}}_a:=W⋉\hat{L}$

Corollary

If $w\in {\hat{W}}_a$ and $H_{\alpha ,k}\in \mathcal{H}$, then $w{H}_{\alpha ,k}=H_{\beta ,l}$ for some $\beta \in \Phi$, $l\in \mathbf{Z}$, and thus $w{s}_{\alpha ,k}{w}^{-1}=s_{\beta ,l}$.

Theorem

Let $w\in W_a$ have a reduced expression $w=s_1\cdots s_r$, with $s_i\in S_a$. If $\ell (ws)<\ell (w)$ ($s\in S_a$), then there exists an index $1\le i\le r$ for which $ws=s_1\cdots {\hat{s}}_i\cdots s_r$.

Proof. By part (c) of Theorem,

$\mathcal{L}(w)=\{H_1,s_{1}H2,\dots ,s_1\cdots s_{r-1}{H}_r\}.$ Therefore $\mathcal{L}(w^{-1})=w^{-1}\mathcal{L}(w)=\{s_r\cdots s_1{H}_1,s_r\cdots s_2{H}_2,\dots ,s_r{H}_r\}.$

Theorem

The pair $(W_a,S_a)$ is a Coxeter system.