Chapter 5. Coxeter groups
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 11 definitions, 5 theorems, 8 propositions, 3 lemmas, 3 corollaries, 3 examples, 4 constructions and 1 remark.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: Coxeter system, rank, and Coxeter graph
A Coxeter system (W,S) is given by generators S and relations (ss′)^{m(s,s′)}=1 with m(s,s)=1, symmetric m(s,s′)≥2 or ∞. The rank is |S|; the associated labelled graph is the Coxeter graph.
Definition: universal Coxeter group
A universal Coxeter group has m(s,s′)=∞ for all distinct generators. Rank two is the infinite dihedral group.
Example: PGL(2,Z) as a rank-three Coxeter group
The rank-three Coxeter system with m(s_1,s_2)=3, m(s_1,s_3)=2, m(s_2,s_3)=∞ is isomorphic to PGL(2,Z), via the displayed order-two matrix classes.
Proposition: sign character of a Coxeter group
There is a unique epimorphism ε:W→{±1} sending every s∈S to −1; in particular the canonical generators have order two.
Definition: length and reduced expression
ℓ(w) is the least number of simple generators in an expression for w; a minimum expression is reduced. The source records the standard elementary length inequalities and ℓ(w^{-1})=ℓ(w).
Proposition: parity of length
ε(w)=(−1)^{ℓ(w)}. Hence for s∈S, both ℓ(ws) and ℓ(sw) equal ℓ(w)±1.
Construction: geometric bilinear form of a Coxeter system
Let V have basis α_s,s∈S. Define symmetric B by B(α_s,α_s)=1 and B(α_s,α_t)=−cos(π/m(s,t)), using −1 when m=∞.
Construction: simple geometric reflections
For s∈S, σ_s(λ)=λ−2B(α_s,λ)α_s is an involutive linear reflection preserving B.
Proposition: geometric representation
There is a unique homomorphism σ:W→GL(V) sending s↦σ_s; σ(W) preserves B, and the order of ss′ is exactly m(s,s′).
Definition: geometric representation
The homomorphism σ:W→GL(V) is called the geometric representation of W.
Definition: abstract root system and positive/negative roots
The root system Φ={w(α_s) | w∈W,s∈S}. A root is positive (negative) when all coordinates in the basis α_s are nonnegative (nonpositive), giving Φ+,Φ−.
Definition: parabolic subgroup in an abstract Coxeter group
For I⊆S, W_I is the subgroup generated by I; its conjugates are parabolic subgroups.
Theorem: length criterion by root sign
For w∈W,s∈S, ℓ(ws)>ℓ(w) iff w(α_s)>0; equivalently ℓ(ws)<ℓ(w) iff w(α_s)<0.
Corollary: faithfulness of the geometric representation
The geometric representation σ:W→GL(V) is faithful.
Theorem: parabolic subgroup theorem
For I⊆S, (W_I,I) is a Coxeter system with inherited Coxeter matrix; every reduced expression of an element of W_I uses only I, so inherited length agrees; I↦W_I is a lattice isomorphism and S is a minimal generating set.
Proposition: simple reflections permute other positive roots and length counts inversions
For s∈S, s sends α_s to −α_s and permutes Φ^+{α_s}. For any w, ℓ(w) equals the number of positive roots sent negative.
Definition: reflections T and root reflection sα
For root α=w(α_s), define s_α=wsw^{-1}; this is independent of choices, acts as λ↦λ−2B(λ,α)α, and α↦s_α is a bijection Φ^+→T, where T is the set of all reflections in W.
Lemma: conjugacy of root reflections
If β=wα then ws_αw^{-1}=s_β.
Proposition: length criterion for arbitrary reflections
For w∈W and α∈Φ^+, ℓ(ws_α)>ℓ(w) iff wα>0.
Theorem: Strong Exchange Condition
If w=s_1⋯s_r is any expression and t∈T satisfies ℓ(wt)<ℓ(w), then wt is obtained by deleting one factor s_i; when the expression is reduced, the deleted index is unique.
Corollary: Deletion Condition and reduced subexpressions
Every nonreduced expression admits deletion of two factors; iterating gives a reduced expression by omitting an even number of factors.
Definition: Bruhat order
For t∈T write w′→w when w=w′t and ℓ(w)>ℓ(w′); Bruhat order is the transitive closure. It is a partial order with unique minimum 1 and equivalently may be defined using reflections on the left.
Examples: Bruhat order in dihedral and symmetric groups
In finite dihedral groups, distinct lengths are comparable. For S_n the source gives transposition and sorted-prefix criteria for Bruhat order; in finite W, left multiplication by the longest element reverses order.
Proposition: lifting property for Bruhat order
If w′<w and s∈S, then either w′s<w or w′s<ws (or both).
Definition: subexpression of a reduced expression
A subexpression of the fixed reduced word s_1⋯s_r is a product s_{i_1}⋯s_{i_q} with increasing indices, not required to be reduced.
Theorem: Subexpression Property for Bruhat order
For any fixed reduced expression w=s_1⋯s_r, w′≤w in Bruhat order iff w′ is represented by a subexpression of that word.
Corollary: Bruhat order restricts to parabolic subgroups
For I⊆S, the intrinsic Bruhat order of (W_I,I) equals the restriction of the Bruhat order of W.
Lemma: rank-one Bruhat diamond step
If w′<w with ℓ(w)=ℓ(w′)+1 and w′<w′s≠w, then w<ws and w′s<ws.
Proposition: Bruhat order is graded by length
If w′<w, there is a saturated chain w′=w_0<w_1<⋯<w_m=w with ℓ(w_i)=ℓ(w_{i−1})+1. Thus adjacent elements differ in length one.
Definition/result: distinguished parabolic coset representatives
W^I={w | ℓ(ws)>ℓ(w) for all s∈I}. Every w factors uniquely w=uv with u∈W^I,v∈W_I and ℓ(w)=ℓ(u)+ℓ(v); u is the unique minimum-length representative of wW_I.
Definition: Poincaré series of a Coxeter group
W(t)=Σ_{w∈W}t^{ℓ(w)} is the Poincaré series; coefficients are finite because S is finite. For subsets X⊆W define X(t) analogously, and W(t)=W^I(t)W_I(t).
Proposition: alternating parabolic identity and rationality of Poincaré series
The alternating identity over I⊆S involving 1/W_I(t) has right side 0 for infinite W and t^N for finite W of longest length N. Induction on |S| then shows W(t) is an explicitly computable rational function.
Example: Poincaré series of the infinite dihedral group
For I_2(∞), W(t)=1+2t+2t^2+⋯=(1+t)/(1−t).
Construction: dual chambers, faces, and the Tits cone
Under the contragredient action on V*, define walls Z_s={f | (f,α_s)=0}, positive halfspaces A_s, chamber C=∩A_s, closed cone D=cl C, faces C_I, and U=∪_{w∈W}wD, the Tits cone.
Lemma: length detects the side of a dual wall
For s∈S,w∈W, ℓ(sw)>ℓ(w) iff w(C) lies in the positive halfspace A_s, and ℓ(sw)<ℓ(w) iff w(C) lies in the negative halfspace.
Theorem: Tits-cone fundamental-domain theorem
If w(C_I) meets C_J, then I=J and w∈W_I. Thus W_I is exactly the stabilizer of C_I; D is a fundamental domain for W on U; U is convex, and any closed segment in U meets only finitely many faces w(C_I).
Remark/comparison: word problem via the faithful Tits-cone action
Choosing f in the open chamber C, no nonidentity w fixes f. Hence equality of a word to 1 can be decided by applying the corresponding dual geometric-representation matrices to f.
Construction/comparison: abstract Coxeter complex and buildings interface
The family of parabolic subgroups yields the abstract Coxeter complex as in the finite case; this is the basic simplicial ingredient in Tits buildings.