Galleries

Let R = (V, φ) be an irreducible reduced root system with positive roots φ+ and Weyl group W. Denote by Q ∨ the group generated by the coroots and by Q∨+ ⊂ Q ∨ the dominant cone. The affine Weyl group is the semidirect product W a = W n Q∨. It acts on V ∗ by affine transformations. For λ ∈ Q ∨ denote by tλ ∈ W a the associated translation. Let Ha be the union of all reflection hyperplanes of reflections in W a. Then Ha = α∈φ+,m∈ZHα,m, where Hα,m = {x ∈ V ∗ | 〈α, x 〉 = m}. Let H±α,m be the associated affine half spaces. The closures of the connected components of V ∗ \ Ha are the alcoves in V ∗. They are fun-damental domains for the W a-action on V ∗. Denote by A the set of all alcoves. The fun-damental alcove is Af = {x ∈ V ∗|0 ≤ 〈α, x 〉 ≤ 1∀α ∈ φ+} ∈ A. We have a bijection W a → A, w 7 → Aw: = wAf. Every alcove A is of the form µ + Aw for unique µ ∈ Q ∨ and w ∈W. Denote this by µ(A) = µ and w(A) = w. A face F of an alcove A is an intersection F = A ∩ H such that H ⊂ Ha is a reflection hyperplane and 〈F 〉aff = H. Here 〈F 〉aff is the affine subspace spanned by F. A wall of A is some hyperplane H ⊂ Ha such that H ∩A is a face of A. The group W a is generated by th