Chapter 6. Special cases
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 5 definitions, 3 theorems, 5 propositions, 2 lemmas, 1 corollary, 2 constructions, 1 remark and 3 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: irreducible Coxeter system
A Coxeter system is irreducible when its Coxeter graph is connected.
Proposition: decomposition into irreducible Coxeter systems
If the Coxeter graph has connected components with vertex sets S_1,…,S_r, then W is the direct product W_{S_1}×⋯×W_{S_r}, with each factor irreducible.
Proposition: discreteness of the geometric representation
The faithful geometric representation σ(W) is a discrete subgroup of GL(V).
Corollary: positive-definite geometric form implies finite W
If the Coxeter bilinear form B is positive definite, then W is finite.
Definition: radical of the Coxeter bilinear form
V^⊥={λ∈V | B(λ,μ)=0 for all μ∈V} is the radical of B. It is W-stable and fixed pointwise by W.
Proposition: invariant subspaces in an irreducible Coxeter representation
For irreducible (W,S): every proper W-stable subspace is contained in rad B; if B is degenerate the geometric module is not completely reducible; if B is nondegenerate it is irreducible; and its commuting endomorphism algebra consists only of scalars.
Lemma: finite real group representations
For a finite group representation on finite-dimensional real E: averaging gives a positive-definite invariant symmetric form; the module is completely reducible; and if the commutant is scalar, any two nondegenerate invariant symmetric forms are scalar multiples.
Theorem: characterization of finite Coxeter groups
The following are equivalent: W is finite; the geometric bilinear form B is positive definite; W in its geometric representation is a finite reflection group in the sense of Part I.
Construction/comparison: affine Coxeter representation as an affine Weyl action
For an irreducible positive-semidefinite non-positive-definite Coxeter graph, rad B is one-dimensional, fixed by W. In the dual, the affine hyperplane (f,λ)=1 carries a Euclidean structure on which the simple generators act as affine reflections, recovering the affine-Weyl configuration and fundamental alcove of Chapter 4.
Definition: crystallographic Coxeter group relative to the geometric representation
W is crystallographic relative to σ if the geometric representation stabilizes a lattice in V.
Necessary condition: edge labels and circuit parity for crystallographic Coxeter groups
Crystallographicity forces m(s,t)∈{2,3,4,6,∞}; moreover every circuit in the Coxeter graph must contain an even number of label-4 edges and an even number of label-6 edges.
Construction: root-rescaling lattice for crystallographic Coxeter groups
Scalars c_s can be chosen so λ_s=c_sα_s span a W-stable lattice when along labels 3,4,6,∞ the ratios are respectively 1,√2 or 1/√2,√3 or 1/√3,1. Circuit parity is exactly the consistency condition.
Proposition: criterion for crystallographicity of a Coxeter group
If the Coxeter graph is a tree, W is crystallographic relative to σ iff all labels lie in {2,3,4,6,∞}. With circuits, this label condition plus even parity of label-4 and label-6 edges on every circuit is necessary and sufficient.
Remark/comparison: Kac–Moody crystallographic convention
The common literature convention calls a Coxeter group crystallographic whenever all labels lie in {2,3,4,6,∞}; these are exactly Weyl groups of Kac–Moody Lie algebras from generalized Cartan matrices, where m_ij=2,3,4,6,∞ according as a_ij a_ji=0,1,2,3,≥4.
Classification/result: rank-three Coxeter signatures
For connected rank-three graphs, finite types give positive-definite B, affine types positive semidefinite B, and all other cases signature (2,1). For triangular labels m,n,p, the sum 1/m+1/n+1/p is >1,=1,<1 exactly in the spherical, Euclidean, hyperbolic cases.
Definition: hyperbolic Coxeter system
An irreducible Coxeter system of rank n is hyperbolic when B has signature (n−1,1) and B(λ,λ)<0 for every λ in the open Tits chamber C.
Lemma: hyperplane restriction criterion in signature (n−1,1)
For nonzero λ in a real quadratic space of signature (n−1,1), the restriction of B to λ^⊥ is of positive type iff B(λ,λ)≤0 (with positive definiteness when <0).
Proposition: graph criterion for hyperbolic Coxeter systems
Irreducible (W,S) is hyperbolic iff B is nondegenerate but not positive definite and deleting any one vertex from the Coxeter graph leaves a graph of positive type.
Theorem/recorded result: finite covolume criterion for Coxeter lattices
For nondegenerate B, the discrete group W≤O(V) has finite Haar covolume iff B is positive definite or W is hyperbolic in the above sense.
Definition: uniform/compact hyperbolic Coxeter group
A hyperbolic Coxeter lattice is uniform/co-compact when O(V)/W is compact.
Theorem/recorded result: compact-hyperbolic graph criterion
W is compact hyperbolic iff B is nondegenerate but not positive definite and deleting any one Coxeter-graph vertex yields a positive-definite graph.
Classification/result: rank bounds for hyperbolic Coxeter groups
Hyperbolic Coxeter systems of simplex type exist only in ranks 3 through 10 and only finitely many in each rank 4–10; compact hyperbolic systems exist only in ranks 3,4,5. Figures 2–3 list the rank≥4 cases.