Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group

Let $E$ be a finite-dimensional real inner-product space with inner product $B$ (Real and complex inner product spaces, with the inner product linear in the first argument). Equip it with the induced norm $\mathrm{\parallel }x{\mathrm{\parallel }}_B=\sqrt{B(x,x)}$ (The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The induced length is a norm) and metric $d_B(x,y)=\mathrm{\parallel }x-y{\mathrm{\parallel }}_B$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms); $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(E)$ below means the isometry group for $d_B$ (Isometry, isometric embedding, and the subspace metric on a subset). Let $\mathrm{\Phi }\subseteq E$ be a supplied reduced crystallographic root system spanning $E$ (Reduced crystallographic Euclidean root system). For each $\alpha \in \mathrm{\Phi }$, put ${\alpha }^{\vee }=2\alpha \mathrm{/}B(\alpha ,\alpha )$ (Coroot and dual root system); write $s_{\alpha }$ for the orthogonal reflection in the hyperplane $B(x,\alpha )=0$ and $W=W(\mathrm{\Phi })=⟨s_{\alpha }:\alpha \in \mathrm{\Phi }⟩$ for the Weyl group (Weyl group). Write $Q={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}\alpha$ and $Q^{\vee }={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}{\alpha }^{\vee }$ for the root and coroot lattices (Root, coroot, weight, and coweight lattices), and fix a positive system with base $\mathrm{\Delta }=\{{\alpha }_s:s\in S\}$ (Positive systems and simple roots).

For $\alpha \in \mathrm{\Phi }$ and $k\in \mathbb{Z}$, define $H_{\alpha ,k}:=\{x\in E:B(x,\alpha )=k\},r_{\alpha ,k}(x):=x-(B(x,\alpha )-k){\alpha }^{\vee }\mathrm{.}$ Each $H_{\alpha ,k}$ is an affine hyperplane: $B(-,\alpha )$ is a nonzero linear functional, and $k\alpha \mathrm{/}B(\alpha ,\alpha )$ lies in its level-$k$ set. The sets $H_{\alpha ,k}$ are the affine root hyperplanes or walls, and ${\mathcal{A}}_{\mathrm{\Phi }}:=\{H_{\alpha ,k}:\alpha \in \mathrm{\Phi },k\in \mathbb{Z}\}$ is their affine wall arrangement. An alcove is a connected component of $E\setminus \bigcup {\mathcal{A}}_{\mathrm{\Phi }}$, with the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Connected components, quasicomponents, and totally disconnected spaces).

The affine reflection group $W_a$ is the subgroup of the Euclidean isometry group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(E)$ generated by the maps $r_{\alpha ,k}$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups, Isometry, isometric embedding, and the subspace metric on a subset). The well-definedness obligation for this subgroup is assigned to lem-cg-affine-reflection-identities-and-local-finiteness, which verifies that each generator is an isometry. The coroot translation group is the image of the homomorphism $Q^{\vee }\to \mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(E)$, $\lambda \mapsto t_{\lambda }$, where $t_{\lambda }(x)=x+\lambda$.

The Weyl group acts on $Q^{\vee }$ by $w\cdot \lambda :=w(\lambda )$. This action is by automorphisms: $W$ permutes $\mathrm{\Phi }$, and for every root $\alpha$, $w({\alpha }^{\vee })=\frac{2w\alpha }{B(\alpha ,\alpha )}=\frac{2w\alpha }{B(w\alpha ,w\alpha )}=(w\alpha {)}^{\vee },$ so $w(Q^{\vee })=Q^{\vee }$ and the restriction has inverse $w^{-1}{\mathrm{\mid }}_{Q^{\vee }}$. These restrictions compose according to multiplication in $W$, giving the action required for the external semidirect product $Q^{\vee }⋊W$ (Left group actions, transitive actions, and faithful actions, An action of a group $H$ on a group $N$ by automorphisms, The external semidirect product $N{⋊}_{\alpha }H$), whose multiplication is $(\lambda ,w)(\mu ,v)=(\lambda +w\mu ,wv)\mathrm{.}$

This item fixes the definitions and conventions. It does not assert $W_a=Q^{\vee }⋊W$, local finiteness of the arrangement, that alcoves are simplices, or that any specified alcove is a fundamental domain. The root system is supplied; no crystallographic scaling for a Coxeter matrix is constructed, and non-crystallographic types are outside this definition. No choice principle is used.