Lecture 15 - Root System Axiomatics

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1 Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the origin. about this hyperplane, denoted σ v, is just We make the general definition meaning σ v (x) = x w, v (x, v) (v, v) (w, v) (v, v) Reflection v. (1) σ v (x) = x x, v v (3) Lemma 1.1 Let Φ R n be a finite set which spans R n, and also has the property that v Φ implies that σ v leaves Φ invariant. Assume σ GL(R n ) leaves Φ invariant, fixes a hyperplane P R n in the pointwise sense, and sends some vector v R n to its negative. Then σ = σ v and P = P v. () Pf. Set τ = σ σ v, so obviously τ : Φ Φ. Note that τ : Rv Rv acts as the identity. We can identify R n P Rv so w R n is w = w P + w v, w P P and w v Rv. We have σ v (w P + w v ) = w P (w P,v) (v,v) v w v so τ(w P ) = w P + (w P,v) (v,v) v + w v. Thus τ acts on R n /Rv P, and acts as the identity. This means τ has only unity eigenvalues, so its minimal polynomial divides (T 1) n. If w Φ then {w, τ(w),..., τ k (w)} cannot all be independent if k Card(Φ). Thus there is some number l so that τ l acts as the identity on Φ and therefore on R n (l will be the l.c.m. of the various k for the various w Φ). Thus also the minimal polynomial of τ divides T l 1. However gcd ( T k 1, (T 1) n) = T 1, so the minimal polynomial is T 1. Thus τ = Id. 1

2 Root System Axiomatics We say a subset Φ of E = R n (with its Euclidean metric) is a root system provided R1) Φ is finite, spans E, and does not contain 0 R) If α Φ, then cα Φ if only if c = ±1 R3) If α Φ, then σ α leaves Φ invariant R4) If α, β Φ, then β, α Z The dimension of R n is called the rank of the root system. Given a root system Φ E, denote by W GL(E) the group of transformations generated by σ α for all α Φ. This is called the Weyl group for the root system. Lemma.1 If σ GL(E) leaves Φ invariant, then σσ α σ 1 = σ σ(α) and β, α = σ(β), σ(α) for all α, β Φ. Pf. There is some γ Φ so that σ 1 (β) = γ. Thus σσ α σ 1 (β) = σσ α (γ) = σ (γ γ, α α) (4) and since γ γ, α α is a root, so is σ(γ γ, α α). Thus σσ α σ 1 leaves Φ invariant. Also, σσ α σ 1 sends σ(α) to σ(α), and fixes the plane σ(p α ). By the lemma we have σσ α σ 1 = σ σ(α). For the second assertion, σ (β σ(β), σ(α) α) = σ(β) σ(β), σ(α) σ(α) = σ σ(α) (σ(β)) = σσ α σ 1 σ(β) = σσ α (β) = σ (β β, α α). Taking σ 1 we get the assertion. (5) If α Φ define its dual ˆα to be ˆα = α (α, α) (6) and define ˆΦ = {ˆα α Φ}. This is also a root system, and the Weyl groups of Φ and ˆΦ are canonically isomorphic. In the Lie algebra situation, ˆα is the metric dual (under the Killing form) to h α.

3 Proposition. Assume α, β Φ. We have α, β β, α 0. The only possibilities for α, β are 0, ±1. ±, ±3. If θ is the angle between α and β, then (up to sign) θ can only take on the values 0, π 6, π 4, π 3, π, π 3, 3π 4, 5π 6. Finally, one of α, β, β, α is ±1. Pf. Since cos θ = (α,β) α β so α, β β, α = 4 cos (θ). (7) Since 0 4 cos θ 4, and because 4 cos θ = 4 only when θ = 0, π, the first two assertions hold. Obviously cos θ = 0, ±1, ± 1, ± 3, so the third assertion also holds. The final assertion is obvious from the fact that θ π. Proposition.3 Let α, β Φ be non-proportional roots. If (α, β) > 0 (the angle is strictly acute) then α β is a root. If (α, β) < 0 (the angle is strictly obtuse) then α + β is a root. Pf. The second assertion follows from the first, by replacing α by α. Both α, β, β, α are positive, so by the previous proposition, one of them equals 1. If α, β = 1 then is in Φ. If β, α = 1 then σ β (α) = α α, β β = α β (8) σ α (β) = β β, α α = β α (9) is in Φ, so also α β Φ. The set of all roots of the form β + iα for i Z is called the α-string through β. Proposition.4 Facts about strings: The α-string through β is unbroken. σ α reverses all α-strings. Strings have length at most 4. Pf. Let q, r Z 0 be integers so β rβ Φ but β r α Φ for any r > r, and so β + qα Φ by β + q α Φ for any q > q. Now assume p, s can be found with r p < p + 1 s 1 < s q so that β + pα Φ and β + sα Φ, by β + p α Φ for all p + 1 p s 1. Then according to the lemma, we have both (α, β) + p α = (α, β + pα) > 0 and (α, β) + s α = (α, β + sα) < 0. Thus p < (α,β) α < s which is impossible by assumption. 3

4 so that Because σ α reverses the string, we have σ α (β + qα) = β rα (10) Therefore q r is at most 4. β β, α α qα = β rα β, α α = (q r)α (11) 3 Bases and the Weyl group We have seen that new roots can be constructed from old, via strings. The question naturally arises, can we find some minimal set of roots that generates all the others? A subset Φ is called a base if B1) is a basis of E B) Each root β can be written as a sum β = c α α where α and k α Z where all k α are either all positive or all negative (or zero). If a base for Φ exists, we can say a root is positive or negative if its coefficients are all positive or negative. We can also define the height of a root by htβ = k α where β = k α α. Of course if a root system has more than one base, a root might have different heights relative to these bases. However because any given base is a basis, a root has a well-defined height relative to a given base. Finally we have an ordering of roots. Given roots β, α we have α > β (1) provided α β is a root and is positive. Obviously this is only a partial ordering. Lemma 3.1 If is a base of Φ, then (α, β) 0 for α β for all α, β, and α β is not a root. Pf. Easy. Let γ E be any vector. We call γ regular if it does not lie in P α for any α Φ. Define Φ + (γ) to be Φ + (γ) = { α Φ (α, γ) > 0 }. (13) Clearly if γ is regular, then Φ = Φ + (γ) Φ + (γ). An element α Φ is called indecomposable provided α = β 1 + β implies that either β 1 or β is not in Φ + (γ). 4

5 Theorem 3. Let γ E be regular. Then the set (γ) of indecomposable roots in Φ + (γ) is a base. Further, every base is a (γ) for some regular γ. Pf. Next time. 5

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