Chapter 4. Affine reflection groups
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 11 definitions, 8 theorems, 4 propositions, 1 lemma, 2 corollaries, 2 examples, 1 construction and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: affine group and translations
Aff(V)=GL(V)⋉V is the affine group. Translation by λ is t(λ):μ↦μ+λ, and g t(λ) g^{-1}=t(gλ).
Definition: affine root hyperplanes and affine reflections
For root α and k∈Z, H_{α,k}={λ | (λ,α)=k}; the corresponding affine reflection is s_{α,k}(λ)=λ−((λ,α)−k)α∨=t(kα∨)s_α.
Proposition: affine hyperplanes under Weyl transformations and translations
For w∈W, wH_{α,k}=H_{wα,k} and ws_{α,k}w^{-1}=s_{wα,k}. If λ pairs integrally with all roots, then t(λ)H_{α,k}=H_{α,k+(λ,α)} and conjugates the corresponding reflections accordingly.
Definition: affine Weyl group
The affine Weyl group W_a is the subgroup of Aff(V) generated by all s_{α,k}, α∈Φ,k∈Z.
Example: affine type A1 and the infinite dihedral group
For |W|=2, W_a is generated by s_α and s_{α,1} with only the involution relations; it is the infinite dihedral group I_2(∞), affine type ilde A_1.
Definition: coroot and coweight lattices used by affine Weyl groups
Let L=L(Φ∨) be the coroot lattice and L=L^*(Φ∨) the coweight lattice, characterized by integral pairings with all roots. W stabilizes both lattices.
Proposition: semidirect-product structure of Wa
W_a=W⋉L, where L=L(Φ∨) acts by translations.
Definition/result: extended affine Weyl group
The extended affine Weyl group W_a=W⋉L contains W_a as a normal subgroup of finite index, with quotient W_a/W_a≅L/L.
Corollary: affine hyperplanes form a Wa-stable arrangement
Every w∈W_a sends an affine root hyperplane H_{α,k} to some H_{β,l}, and conjugates its affine reflection to s_{β,l}.
Definition: alcove
Alcoves are the connected components of V minus the union of all affine root hyperplanes.
Definition: fundamental alcove A0
For irreducible Φ with simple roots Δ and positive roots Φ^+, A_0={λ | 0<(λ,α)<1 for all α∈Φ^+}; equivalently 0<(λ,α_i) for simple roots and (λ, ilde α)<1 for the highest root ilde α. It is an open Euclidean simplex.
Examples: rank-two affine alcoves
For A_2,B_2,G_2, the affine alcoves are triangles with angles (π/3,π/3,π/3), (π/2,π/4,π/4), and (π/2,π/3,π/6), respectively.
Definition: walls and simple affine reflections Sa
The walls of A_0 are H_α for α∈Δ and H_{ ilde α,1}; S_a consists of their reflections {s_α | α∈Δ}∪{s_{ ilde α,1}}.
Proposition: transitivity on alcoves and generation by wall reflections
W_a acts transitively on the alcoves and is generated by S_a, the reflections in the walls of A_0.
Definition: affine length and reduced expressions
For w∈W_a, ℓ(w) is the least number of factors from S_a in an expression for w; a minimum-length expression is reduced.
Definition: separating-hyperplane set C(w) and n(w)
C(w) is the finite set of affine root hyperplanes separating A_0 from wA_0; n(w)=|C(w)|. One has n(1)=0, n(s)=1 for s∈S_a, and n(w^{-1})=n(w).
Proposition: change of n(w) under a simple affine reflection
For w∈W_a,s∈S_a, the wall H_s belongs to exactly one of C(w{-1}),C(sw{-1}); reflection in H_s pairs all other separating hyperplanes. Consequently n(ws)=n(w)±1 with the sign determined by membership of H_s in C(w^{-1}).
Corollary: hyperplane number bounded by affine length
For every w∈W_a, n(w)≤ℓ(w).
Lemma: walls arising from a reduced affine expression are distinct
If w=s_1⋯s_r is reduced, the hyperplanes H_1,s_1H_2,…,s_1⋯s_{r−1}H_r are pairwise distinct.
Theorem: affine length equals the number of separating hyperplanes
For reduced w=s_1⋯s_r, C(w) is exactly {H_1,s_1H_2,…,s_1⋯s_{r−1}H_r}; hence n(w)=ℓ(w).
Theorem: simple transitivity on alcoves
W_a acts simply transitively on the set of alcoves.
Definition/result: stabilizer Ω in the extended affine Weyl group
Let Ω≤W_a be the stabilizer of A_0. Then W_a=W_a⋊Ω, W_a∩Ω=1, and Ω≅W_a/W_a≅L/L.
Theorem/result: affine Exchange and Deletion Conditions
If reduced w=s_1⋯s_r and s∈S_a satisfies ℓ(ws)<ℓ(w), then ws is obtained by deleting one factor s_i. Thus the affine Exchange Condition and equivalent Deletion Condition hold.
Theorem: affine Weyl group is a Coxeter group
The pair (W_a,S_a) is a Coxeter system.
Construction/comparison: affine Coxeter graphs and extended Dynkin diagrams
The Coxeter graph of W_a is obtained by adjoining the highest-root affine vertex to the finite Coxeter graph, yielding the standard affine diagrams ilde A_n,…, ilde G_2. Orienting multiple edges using − ilde α gives the extended Dynkin diagram.
Theorem: fundamental domain for Wa
The closure of the fundamental alcove A_0 is a fundamental domain for the action of W_a on V; its stabilizers are generated by the wall reflections fixing the point.
Theorem: Weyl-group order from the highest-root coefficients
If irreducible W has rank n, highest root ilde α=Σ c_i α_i, and f=[L:L] is the index of connection, then |W|=n! c_1⋯c_n f.
Comparison/table: highest-root coefficients and indices of connection
The source tabulates the coefficients c_i and index f for every irreducible Weyl type A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2, supporting the order formula.
Theorem/recorded result: general discrete affine reflection groups are Coxeter groups
For a locally finite/proper discrete affine reflection arrangement, the group G generated by wall reflections acts simply transitively on chambers, is generated by the finitely many wall reflections of a fixed chamber, and with these generators is a Coxeter system.
Theorem/recorded classification: irreducible infinite discrete affine reflection groups are affine Weyl groups
An infinite irreducible discrete group generated by affine reflections contains a full-rank translation lattice equal to the root lattice of a crystallographic root system; the stabilizer of a special point is its finite Weyl group. Hence every such group is an affine Weyl group.