Chapter 1. Finite reflection groups
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 19 definitions, 9 theorems, 8 propositions, 3 lemmas, 4 corollaries, 4 examples, 3 constructions and 1 remark.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: reflection in a Euclidean space
For nonzero a in Euclidean V, the reflection s_a fixes H_a=a^⊥ pointwise and sends a to −a; explicitly s_a(λ)=λ−2(λ,a)/(a,a)a. It is orthogonal, has order two, and depends only on the line Ra.
Definition: finite reflection group
A finite reflection group is a finite subgroup of O(V) generated by reflections.
Example: dihedral reflection group I2(m)
The symmetry group of a regular m-gon is the dihedral group of order 2m, generated by two adjacent reflections; their product is rotation through 2π/m.
Example: symmetric group as type A{n-1}
S_n acts orthogonally on R^n by permuting basis vectors; transpositions are reflections, adjacent transpositions generate, and restriction to the coordinate-sum-zero hyperplane gives the essential type A_{n−1} action.
Definition: essential reflection-group action
A reflection group W acting on V is essential relative to V if its fixed-point space is zero. Any orthogonal action restricts to an essential action on the orthogonal complement of its fixed-point space.
Example: type Bn signed permutation group
Permutations together with independent coordinate sign changes form the reflection group of type B_n, of order 2^n n!, acting essentially on R^n.
Example: type Dn even signed permutation group
Permutations together with even coordinate sign changes form the type D_n reflection group, an index-two subgroup of type B_n, generated by appropriate reflections and acting essentially.
Proposition: orthogonal conjugacy of reflections
For t∈O(V) and 0≠a∈V, t s_a t^{-1}=s_{ta}. Hence a finite reflection group permutes the root lines of its reflections.
Definition: root system for a finite reflection group
A root system Φ⊂V is a finite set of nonzero vectors satisfying Φ∩Rα={α,−α} and s_αΦ=Φ for all α∈Φ. Its associated reflection group is generated by the s_α. Any such group is finite.
Remark/comparison: normalization of a root system
Replacing every root line by its two unit vectors produces another root system with the same associated reflection group.
Definition: total ordering and positive vectors
A total ordering on V is compatible with addition and multiplication by positive scalars, with sign reversal for negative scalars. A vector is positive if it is greater than zero; lexicographic order from an ordered basis supplies such an ordering.
Definition: positive and negative root systems
A positive system Π consists of the roots positive for some total ordering; Φ is the disjoint union Π⊔(−Π), with −Π the corresponding negative system.
Definition: simple system and simple roots
A simple system Δ⊂Φ is a basis of span_R Φ such that every root has coordinates in Δ all nonnegative or all nonpositive. Elements of Δ are simple roots.
Theorem: existence and uniqueness of positive/simple systems
A simple system lies in a unique positive system; every positive system contains a unique simple system. In particular simple systems exist.
Corollary: simple roots have nonpositive mutual inner products
If α≠β are simple roots, then (α,β)≤0.
Definition: rank of a finite reflection group
The rank of W is the cardinality of a simple system, equivalently dim span_R Φ.
Proposition: simple reflection permutes the other positive roots
If α∈Δ⊂Π, then s_α maps Π minus {α} bijectively to itself; α is the unique positive root made negative by s_α.
Theorem: conjugacy of positive and simple systems
Any two positive systems, and hence any two simple systems, are conjugate under W.
Definition: height of a root
For β=Σ_{α∈Δ}c_α α, the height ht(β)=Σ c_α relative to Δ.
Theorem: generation by simple reflections
For a fixed simple system Δ, W is generated by the simple reflections s_α, α∈Δ.
Corollary: every root is conjugate to a simple root
For every β∈Φ there exists w∈W with wβ∈Δ.
Definition: length and reduced expression
Relative to the simple-reflection set S, ℓ(w) is the minimum length of an expression w=s_1⋯s_r with s_i∈S; an expression of minimum length is reduced. One has ℓ(1)=0, ℓ(w^{-1})=ℓ(w), and det(w)=(−1)^{ℓ(w)}.
Definition: inversion number n(w)
For the fixed positive system Π, n(w)=|Π∩w^{-1}(−Π)| is the number of positive roots sent negative. One has n(w^{-1})=n(w).
Lemma: change of inversion number under a simple reflection
For α∈Δ: if wα>0 then n(ws_α)=n(w)+1, while if wα<0 then n(ws_α)=n(w)−1; analogous formulas hold for multiplication on the left.
Corollary: inversion number bounds expression length
If w is expressed as a product of r simple reflections, n(w)≤r; in particular n(w)≤ℓ(w).
Theorem: Deletion Theorem
If w=s_1⋯s_r is an expression in simple reflections with n(w)<r, then two factors can be deleted without changing w; the proof gives the corresponding root identity and factor-exchange identity.
Corollary: length equals inversion number
For all w∈W, ℓ(w)=n(w).
Definition/result: Deletion Condition
Every nonreduced expression in simple reflections admits deletion of two factors without changing the represented element; repeated deletions produce a reduced expression.
Construction: inversion set from a reduced expression
For reduced w=s_1⋯s_r with s_i=s_{α_i}, the inversion set Π(w)=Π∩w^{-1}(−Π) is exactly {β_i=s_r⋯s_{i+1}(α_i) | 1≤i≤r}; the β_i are distinct.
Theorem/result: Exchange Condition
If w=s_1⋯s_r and s is simple with ℓ(ws)<ℓ(w), then ws is obtained by deleting one factor s_i from the displayed expression. In particular a reduced expression for w can end in s iff ℓ(ws)<ℓ(w).
Theorem: simple transitivity on positive/simple systems
For w∈W the conditions wΠ=Π, wΔ=Δ, n(w)=0, ℓ(w)=0, and w=1 are equivalent. Thus W acts simply transitively on positive systems and on simple systems.
Theorem/result: longest element
There is a unique w_0 sending Π to −Π. It has maximal length |Π|, satisfies w_0^2=1 and ℓ(w_0w)=ℓ(w_0)−ℓ(w); every w occurs as a length-additive prefix of w_0.
Theorem: Coxeter presentation of a finite reflection group
For simple reflections S={s_α}, W is generated by S subject only to (s_αs_β)^{m(α,β)}=1, where m(α,β) is the order of s_αs_β.
Definition: Coxeter group and Coxeter system
A Coxeter system (W,S) is a group with generators S and relations (ss′)^{m(s,s′)}=1, where m(s,s)=1 and m(s,s′) may be ∞, in which case the relation is omitted. W is then a Coxeter group.
Definition: standard parabolic subgroup
For I⊆S, W_I is the subgroup generated by I. Subgroups obtained from these by conjugacy are called parabolic subgroups.
Proposition: root systems and minimal representatives for parabolic subgroups
For I⊆S, Φ_I=Φ∩span Δ_I is a root system with simple system Δ_I and reflection group W_I; length restricts from W to W_I. Every w factors uniquely w=uv with u∈W^I={x | ℓ(xs)>ℓ(x) for all s∈I}, v∈W_I, and ℓ(w)=ℓ(u)+ℓ(v); u is the unique minimum-length representative of wW_I.
Definition: minimal coset representative
The elements W^I in the parabolic decomposition are the distinguished/minimal coset representatives for W/W_I.
Definition: Poincaré polynomial
If a_n=|{w∈W | ℓ(w)=n}|, the Poincaré polynomial is W(t)=Σ_{n≥0}a_ntn=Σ_{w∈W}t{ℓ(w)}. More generally X(t)=Σ_{w∈X}t^{ℓ(w)} for X⊆W.
Proposition: parabolic factorization and alternating formula for W(t)
For I⊆S, W(t)=W^I(t)W_I(t). If N=ℓ(w_0), the source derives the alternating subset identity Σ_{I⊆S}(−1){|I|}WI(t)=t^N and its specialization at t=1 (Witt’s identity).
Definition/construction: fundamental chamber and closed cone
For Δ⊂Π, put C=⋂_{α∈Δ}{λ | (λ,α)>0} and D=cl(C)={λ | (λ,α)≥0 for all α∈Δ}. These are open/closed convex cones.
Lemma: existence of a dominant orbit representative
Every λ∈V is W-conjugate to some μ∈D, and μ−λ is a nonnegative real linear combination of simple roots.
Theorem: fundamental-domain and isotropy theorem
If wλ=μ with λ,μ∈D, then λ=μ and w is a product of simple reflections fixing λ. Hence D is a fundamental domain, interior points have trivial isotropy, and the stabilizer of any vector—or pointwise stabilizer of any subset—is generated by the reflections it contains.
Definition: chambers and walls
The chambers are the connected components of V minus the union of reflecting hyperplanes, equivalently the cones wC. Walls of C are the simple-root hyperplanes; W acts simply transitively on chambers.
Proposition: lattice of standard parabolic subgroups
The correspondence I↦W_I is a lattice isomorphism from subsets of S to standard parabolic subgroups; in particular W_I∩W_J=W_{I∩J} and the subgroup generated by W_I,W_J is W_{I∪J}.
Proposition: all reflections are root reflections
Every reflection contained in W equals s_α for some α∈Φ.
Definition/construction: facets and the Coxeter complex
For I⊆S, C_I is the stratum of D on which simple-root pairings vanish precisely for Δ_I. The sets wC_I partition V; their collection is the Coxeter complex, and wC_I is a facet of type I.
Proposition: parabolic subgroups as facet stabilizers
The isotropy group of C_I is exactly W_I; hence standard parabolic subgroups are precisely stabilizers of facets of the Coxeter complex.
Construction/comparison: abstract and geometric Coxeter complex
Abstractly, vertices are cosets wW_I for maximal proper I⊂S and simplices are finite collections with nonempty intersection. If Δ spans V, intersecting facets with the unit sphere realizes a W-equivariant simplicial decomposition of S^{n−1}.
Construction: hyperplane arrangement complex
For finitely many hyperplanes with chosen sides, the nonempty intersections of side/face choices form a complex K; dimension means the dimension of the linear span of a face.
Lemma: Euler alternating-sum formula for a hyperplane complex
If n_i is the number of i-dimensional faces in the arrangement complex in n-dimensional V, then Σ_i(−1)^i n_i=(−1)^n.
Proposition: alternating determinant formula from the Coxeter complex
For w∈W, let f_I(w) be the number of type-I facets fixed by w, equivalently fixed left cosets of W_I. Then det(w)=Σ_{I⊆S}(−1)^{|I|}f_I(w).