LetΦbe a nonempty finite set of nonzero vectors inV

May 30, 2016

For today’s lecture, we letV be a finite-dimensional vector space overR, with positive- definite inner product. Recall that for06=α ∈V,s_α ∈O(V)denotes the reflection

s_α(λ) =λ− 2(λ, α)

(α, α) α (λ∈V).

Lemma 1. Fort∈O(V)and06=α ∈V, we havetsαt⁻¹ =stα.

Definition 2. LetΦbe a nonempty finite set of nonzero vectors inV. We say thatΦis a root systemif

(R1) Φ∩Rα={α,−α}for allα∈Φ, (R2) s_αΦ = Φfor allα∈Φ.

Definition 3. Atotal orderingofV is a transitive relation onV (denoted<) satisfying the following axioms.

(i) For each pairλ, µ∈V, exactly one ofλ < µ,λ=µ,µ < λholds.

(ii) For allλ, µ, ν ∈V,µ < ν impliesλ+µ < λ+ν.

(iii) Letµ < νandc∈R. Ifc >0thencµ < cν, and ifc <0thencν < cµ.

For convenience, we writeλ > µifµ < λ. By (ii),λ >0implies0>−λ. Thus V =V₊∪ {0} ∪V₋ (disjoint),

where

V+ ={λ ∈V |λ >0}, V− ={λ ∈V |λ <0}.

Lemma 4. Let<be a total ordering ofV, and letλ, µ∈V.

(i) Ifλ, µ >0, thenλ+µ >0.

(ii) Ifλ >0,c∈Randc >0, thencλ > 0.

(iii) Ifλ >0,c∈Randc <0, thencλ < 0. In particular,−λ <0.

Lemma 5. Let∆ be a finite set of nonzero vectors inV₊. If (α, β) ≤ 0for any distinct α, β ∈∆, then∆consists of linearly independent vectors.

Lemma 6. Let ∆ ⊂ V₊ be a subset, and let α, β ∈ ∆be linearly independent. If α ∈ R_>0β+R≥0∆, thenα∈R≥0(∆\ {α}).

1

Definition 7. LetΦbe a root system inV. A subsetΠ ofΦis called apositive system if there exists a total ordering<ofV such thatΠ ={α∈Φ|α >0}.

Lemma 8. If Π is a positive system in a root system Φ, then Φ = Π∪(−Π) (disjoint), where

−Π ={−α|α ∈Π}.

In particular,

−Π ={α∈Φ|α <0}.

Definition 9. Let∆ be a subset of a root systemΦ. We call∆asimple system if∆is a basis of the subspace spanned byΦ, and if moreoverΦ⊂R≥0∆∪R≤0∆holds.

In what follows, we fix a root systemΦ inV. Recall thatP(Φ) andS(Φ)denote the set of positive systems and that of simple systems, respectively, inΦ.

Lemma 10. If∆∈ S(Φ),Π∈ P(Φ)and∆⊂Π, then

(i) Π = Φ∩R≥0∆,

(ii) ∆ = {α∈Π|α /∈R_≥0(Π\ {α})}.

Theorem 11. The mapping

π :S(Φ) → P(Φ)

∆ 7→ Φ∩R≥0∆ is a bijection whose inverse is given by

π⁻¹ :P(Φ) → S(Φ)

Π 7→ {α∈Π|α /∈R≥0(Π\ {α})}. (1) Moreover,

(i) for every simple system∆inΦ,π(∆)is the unique positive system containing∆, (ii) for every positive systemΠinΦ,π⁻¹(Π)is the unique simple system contained inΠ.

Example 12. Letε₁, . . . , ε_nbe the standard basis ofRⁿ. The set Φ ={±(ε_i−ε_j)|1≤i < j ≤n}

is a root system, with a positive system

Π ={ε_i−ε_j |1≤i < j ≤n}, and simple system

∆ ={ε_i−ε_i+1 |1≤i < n}.

2