Chapter III. Root Systems
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 22 definitions, 7 theorems, 3 propositions, 19 lemmas, 3 corollaries, 1 example, 9 constructions, 1 convention and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: Euclidean ambient space
E is a finite-dimensional real vector space with a positive-definite symmetric bilinear form (·,·).
Definition: reflection and Cartan pairing notation
For 0≠α∈E, σ_α is the orthogonal reflection fixing P_α=α^⊥ and sending α to −α: σ_α(β)=β−⟨β,α⟩α, where ⟨β,α⟩=2(β,α)/(α,α).
Lemma: recognition of a root reflection
If a finite spanning set Φ is invariant under all its root reflections and τ∈GL(E) preserves Φ, fixes a hyperplane pointwise, and sends α∈Φ to −α, then τ=σ_α and the fixed hyperplane is P_α.
Definition: root system
A root system Φ⊂E is finite, spans E, excludes 0, contains only ±α on each root line, is invariant under every σ_α, and satisfies ⟨β,α⟩∈Z for all α,β∈Φ.
Definition: Weyl group
The Weyl group W of Φ is the subgroup of GL(E) generated by the root reflections σ_α. It acts faithfully on the finite spanning set Φ and hence is finite.
Lemma: automorphisms conjugate root reflections
If τ∈GL(E) preserves Φ, then τσ_ατ^{-1}=σ_{τ(α)} and ⟨β,α⟩=⟨τ(β),τ(α)⟩.
Definition: isomorphism and automorphism of root systems
An isomorphism (Φ,E)→(Φ′,E′) is a vector-space isomorphism carrying Φ onto Φ′ and preserving all Cartan pairings. It induces an isomorphism of Weyl groups. Aut Φ is equivalently the group of linear automorphisms of E preserving Φ.
Construction: dual root system
For α∈Φ set α∨=2α/(α,α). The set Φ∨={α∨ | α∈Φ} is the dual/inverse root system, with canonically isomorphic Weyl group.
Definition: rank of a root system
The rank of Φ is dim E.
Examples: root systems of rank one and two
The source exhibits A_1 in rank one and the rank-two systems A_1×A_1, A_2, B_2, and G_2, with their Weyl-group geometry.
Proposition: possible angles and root-length ratios
For nonproportional roots α,β, the integer product ⟨α,β⟩⟨β,α⟩=4cos²θ is 0,1,2, or 3, yielding only the standard angles π/2, π/3, 2π/3, π/4, 3π/4, π/6, 5π/6 and squared length ratios 1,2,3 and reciprocals.
Lemma: sum/difference criterion for roots
For nonproportional α,β∈Φ: if (α,β)>0 then α−β∈Φ; if (α,β)<0 then α+β∈Φ.
Definition/theorem: root strings
For nonproportional roots α,β, the α-string through β consists of all β+iα∈Φ. It is unbroken from β−rα through β+qα, σ_α reverses it, r−q=⟨β,α⟩, and its length is at most four.
Definition: base, simple roots, height, positivity, and root order
A base Δ⊂Φ is a basis of E such that every root has integral coordinates all ≥0 or all ≤0. Elements of Δ are simple; the coordinate sum is height. This defines positive/negative roots Φ± and the partial order β≤α when α−β is a nonnegative sum of simple roots.
Lemma: distinct simple roots make obtuse angles
If Δ is a base and α≠β∈Δ, then (α,β)≤0 and α−β is not a root.
Theorem: existence of bases
Every root system has a base.
Definition: regular and singular vectors; decomposable positive root
A vector γ is regular when it lies on no root hyperplane and singular otherwise. For regular γ, Φ⁺(γ)={α | (γ,α)>0}; a root in Φ⁺(γ) is decomposable if it is a sum of two roots of Φ⁺(γ).
Theorem: bases from regular vectors
For regular γ, the indecomposable roots in Φ⁺(γ) form a base Δ(γ), and every base arises this way.
Definition: Weyl chambers and fundamental chamber
The connected components of E minus the root hyperplanes are open Weyl chambers. They correspond bijectively to bases; C(Δ)={γ | (γ,α)>0 for all α∈Δ} is the fundamental chamber for Δ.
Lemma: subtracting a simple root from a nonsimple positive root
If α is positive but not simple, then α−β is a positive root for some β∈Δ.
Corollary: simple-root chain decomposition of a positive root
Every positive root can be written α_1+⋯+α_t with α_i simple and every partial sum a root.
Lemma: simple reflection permutes other positive roots
For simple α, σ_α permutes the positive roots other than α.
Corollary: action of a simple reflection on the half-sum/sum of positive roots
If δ is the sum of all positive roots, then σ_α(δ)=δ−2α for simple α (equivalently the half-sum changes by −α).
Lemma: cancellation in words of simple reflections
If a product of simple reflections sends the final simple root negative, then the word admits the standard deletion relation described in Lemma 10.2C.
Corollary: last simple root in a reduced word goes negative
If w=σ_{α_1}⋯σ_{α_t} is a reduced word in simple reflections, then w(α_t)<0.
Theorem: basic Weyl-group action theorem
Relative to a base Δ: W acts transitively on Weyl chambers and bases; every root is W-conjugate to a simple root; W is generated by simple reflections; and the action on bases is simply transitive.
Definition: length and inversion number in W
The length ℓ(w) is the minimal number of simple reflections in an expression for w. The inversion number n(w) is the number of positive roots sent negative.
Lemma: length equals inversion number
For every w∈W, ℓ(w)=n(w).
Lemma: closed fundamental chamber is a fundamental domain
If λ,μ lie in the closure of the fundamental chamber and wλ=μ, then w is a product of simple reflections fixing λ; in particular λ=μ. Thus each W-orbit meets the closed fundamental chamber exactly once.
Definition: irreducible root system
Φ is irreducible if it cannot be partitioned into two nonempty mutually orthogonal subsets; equivalently no base admits such an orthogonal partition.
Lemma: unique maximal root
For irreducible Φ and a fixed base, there is a unique maximal root β in the root order. Its simple-root coefficients are all strictly positive, and (β,α)>0 for each simple α.
Lemma: irreducibility of the Weyl representation
If Φ is irreducible then W acts irreducibly on E; in particular the W-orbit of any root spans E.
Lemma: root lengths in an irreducible root system
An irreducible root system has at most two root lengths, and all roots of a fixed length are W-conjugate.
Definition: long and short roots
When two root lengths occur they are called long and short; if only one occurs, all roots are conventionally called long.
Lemma: maximal root is long
In an irreducible root system with two root lengths, the unique maximal root is long.
Definition: Cartan matrix of a root system
For an ordered base (α_1,…,α_l), the Cartan matrix is (⟨α_i,α_j⟩). Its entries are Cartan integers; the matrix is nonsingular and is independent up to simultaneous reordering of the chosen base.
Proposition: Cartan matrix determines the root system
A bijection between bases preserving all Cartan integers extends uniquely to an isomorphism of root systems. Hence the Cartan matrix determines Φ up to isomorphism.
Construction: recovery of roots from the Cartan matrix
Using root strings recursively by height, the positive roots and all relevant Cartan integers can be reconstructed from the Cartan matrix.
Definition: Coxeter graph
The Coxeter graph has one vertex per simple root and ⟨α_i,α_j⟩⟨α_j,α_i⟩ edges between distinct vertices i,j.
Definition: Dynkin diagram
The Dynkin diagram is the Coxeter graph with each double/triple edge oriented toward the shorter root; it recovers the Cartan integers.
Proposition: decomposition into irreducible components
Every root system decomposes uniquely as a union of irreducible root systems Φ_i in mutually orthogonal subspaces E_i with E=⊕E_i.
Theorem: classification of irreducible root systems
Every irreducible root system has Dynkin diagram of type A_l, B_l, C_l, D_l, E_6,E_7,E_8,F_4, or G_2, with the standard rank restrictions.
Definition: admissible set used in Dynkin classification
An admissible set is a linearly independent set of unit vectors with pairwise nonpositive inner products and 4(ε_i,ε_j)^2∈{0,1,2,3}; its graph is defined as for the Coxeter graph.
Definition: standard lattice in R^n
For the standard orthonormal basis ε_1,…,ε_n of R^n, I denotes its Z-span, a lattice.
Construction: root system of type Al
In the hyperplane of R^{l+1} orthogonal to ε_1+⋯+ε_{l+1}, Φ={ε_i−ε_j | i≠j} is type A_l with base ε_i−ε_{i+1}; W≅S_{l+1}.
Construction: root system of type Bl
In R^l, roots ±ε_i and ±(ε_i±ε_j) form type B_l with standard base; W is the signed permutation group (Z/2Z)^l⋊S_l.
Construction: root system of type Cl
In R^l, roots ±2ε_i and ±(ε_i±ε_j) form type C_l with the standard base; it is dual to B_l and has the same Weyl group.
Construction: root system of type Dl
In R^l, roots ±(ε_i±ε_j), i≠j, form type D_l with standard base; W consists of permutations and even sign changes, (Z/2Z)^{l−1}⋊S_l.
Construction: root systems E6,E7,E8
The source constructs E_8 from an explicit lattice in R^8 and squared-length-two vectors; E_6,E_7 occur as canonical subsystems, with the displayed bases and Cartan data.
Construction: root system F4
The explicit lattice/vector construction in R^4 using roots ±ε_i, ±(ε_i±ε_j), and half-sums yields type F_4 with the displayed base.
Construction: root system G2
In the plane of R^3 orthogonal to ε_1+ε_2+ε_3, the vectors of squared length 2 or 6 in the induced lattice give type G_2 with the displayed base.
Theorem: existence for every Dynkin type
For every Dynkin diagram/Cartan matrix of type A–G, there exists an irreducible root system with that diagram.
Theorem: automorphism group of a root system
For a fixed base Δ, let Γ={τ∈Aut Φ | τ(Δ)=Δ}. Then Aut Φ=Γ⋉W and Γ identifies with the automorphism group of the Dynkin diagram.
Comparison/table: irreducible root-system Weyl data
For every irreducible type A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2, the source tabulates the number of positive roots, the order and abstract structure of the Weyl group, and the Dynkin-diagram automorphism group Γ; in particular the exceptional positive-root counts are 36,63,120,24,6 for E_6,E_7,E_8,F_4,G_2.
Definition: weight lattice and root lattice
The weight lattice Λ={λ∈E | ⟨λ,α⟩∈Z for all α∈Φ}; the root lattice Λ_r is the subgroup generated by Φ. One has Λ_r⊆Λ.
Definition: dominant, strongly dominant, and fundamental dominant weights
Relative to a base Δ, λ∈Λ is dominant when ⟨λ,α⟩≥0 for all α∈Δ, strongly dominant when all are >0. The dual basis λ_i to the coroots 2α_i/(α_i,α_i) consists of fundamental dominant weights and is a Z-basis of Λ.
Definition/theorem: fundamental group of a root system
Λ/Λ_r is finite and is called the fundamental group of Φ. Its order is |det(Cartan matrix)|; the source lists the orders for each irreducible Dynkin type.
Lemma: unique dominant representative of a Weyl orbit
Every weight is W-conjugate to exactly one dominant weight. If λ is dominant then wλ≤λ for all w∈W; if λ is strongly dominant, wλ=λ only for w=1.
Lemma: finiteness below a dominant weight
For λ∈Λ, only finitely many dominant weights μ satisfy μ≤λ.
Definition/theorem: half-sum of positive roots
Let δ=(1/2)Σ_{α>0}α. Then δ=Σ_i λ_i and is strongly dominant; σ_iδ=δ−α_i.
Lemma: norm inequality under Weyl conjugation
If μ is dominant and ν=w^{-1}μ, then (ν+δ,ν+δ)≤(μ+δ,μ+δ), with equality only when ν=μ.
Definition: saturated set of weights and highest weight
A subset Π⊆Λ is saturated if λ∈Π and 0≤i≤⟨λ,α⟩ imply λ−iα∈Π. It is W-stable. A saturated Π has highest weight λ∈Λ⁺ if every μ∈Π satisfies μ≤λ.
Lemma: finite saturated sets with highest weight
Every saturated set of weights with a highest weight is finite.
Lemma: all lower dominant weights occur in a saturated set
If Π is saturated with highest weight λ and μ is dominant with μ≤λ, then μ∈Π.
Theorem: characterization of saturated sets by dominant weights
A saturated set with highest weight λ consists exactly of the dominant weights μ≤λ together with all their W-conjugates; conversely this prescription yields the unique saturated set of highest weight λ.
Lemma: highest-weight norm inequality for saturated sets
If Π is saturated with highest weight λ and μ∈Π, then (μ+δ,μ+δ)≤(λ+δ,λ+δ), with equality only for μ=λ.