ROOT SYSTEMS GEORGE H. SEELINGER 1. Introduction A root system is a configuration of vectors in some Euclidean space that satisfy nice properties concerning reflections [Wik17]. In particular, root systems are acted upon by associated Weyl groups of reflections. While root systems have some interesting features in their own right, they are intimately related to the structure of semisimple Lie algebras. However, this write-up seeks to simply talk about what is known about root systems in their own right. A lot of the definitions and treatments are lifted from [Hum72] and thus I do not lay any claim to the originality of statements of definitions, theorems, etc. I view these notes as mainly a good companion to be read along with [Hum72]. 2. Definitions and Basic Concepts Given a vector α 2 E, we can define a function σα : E ! E that reflects every vector over the hyperplane perpendicular to α. Explicitly, we get 2(β; α) σ (β) = β − α α (α; α) where β 2 E and (·; ·) is the standard inner product on E. To see this works, note that σα(α) = −α as desired and if (β; α) = 0, then σα(β) = β, leaving all the vectors in the hyperplane fixed. 2.1. Definition. For notational convenience, we will say 2(β; α) hβ; αi = : (α; α) It is important to note that h·; ·i is linear only in the first variable. 2.2. Definition. [Hum72, p 42] A root system is a set of vectors Φ in Eu- clidean space E such that • Φ is finite, span Φ = E, and 0 62 Φ. • If α 2 Φ, then the only multiples of α in Φ are ±α. • If α 2 Φ, then σα(Φ) = Φ • If α; β 2 Φ, then hβ; αi 2 Z. Date: February 2017. 1 2 2.3. Example. Consider the collection of vectors in R of equal length with angle from the origin given in multiples of 60 degress. Such a collection is a root system, as can be easily checked. Figure 1. Root system A2 from [Wik17] These hyperplane reflections form a \nice" action of the root system. Furthermore, they form a group, since the composition of two reflections is a reflection and each of these described reflections is their own inverse. Thus, we have the following definition. 2.4. Definition. The Weyl group of a root system is the collection of these reflections σα with group operation as composition. Explicitly, 2 W = hfσα j 8α 2 Φg j fσα = 1; 8α 2 Φgi It is a straightfoward exercise to show the following fact 2.5. Lemma. [Hum72, p 43] If σ 2 GL(E) leaves Φ invariant and σα 2 W is a simple reflection, then −1 σσασ = σσ(α) and also hσ(β); σ(α)i = hβ; αi 2.6. Definition. We say a root system is reducible if there exists root sys- tems Φ1; Φ2 ⊆ Φ such that Φ = Φ1 t Φ2 such that span(Φ) = span(Φ1) ⊕ span(Φ2). Any root system that is not reducible is called irreducible. In general, we restrict our attention to irreducible root systems, since they are the basic building blocks of root systems. Now, we also have a convenience of notation as follows 2.7. Definition. For α 2 Φ, a root system, we say a coroot α_ is given by 2α α_ = (α; α) It is worth noting that Φ_ is also a root system with an associated Weyl group that is isomorphic to the Weyl group of Φ. 2.8. Definition. We define the rank of a root system Φ to be the dimension of Euclidean space it spans. 2 The axioms for a root system are fairly restrictive, and so it is relatively 2 easy to classify all the possible root systems that span R and R . Namely, since (α; β) = kαk kβk cos θ, 2 hα; βihβ; αi = 4 cos θ 2 Z≥0 Thus, we have the following 2.9. Lemma. Let α; β 2 Φ with β 6= ±α, then hα; βihβ; αi 2 f0; 1; 2; 3g Proof. This follows immediately from hα; βi and hβ; αi being integers and 2 4 cos θ ≤ 4. Thus, roots can only ever have certain angles between them. To see this, examine Table 1 on page 45 of [Hum72]. However, these restrictions lead us to a useful application for determining when sums and differences of roots are roots. 2.10. Lemma. [Hum72, p 45] Let α; β 2 Φ be nonproportional roots. • If (α; β) > 0, then α − β 2 Φ. • If (α; β) < 0, then α + β 2 Φ. Proof. Note (α; β) > 0 () hα; βi > 0. By the table of values, the acuteness of the angle requires that either hα; βi or hβ; αi is 1. Thus, by the axioms of the root system, hα; βi = 1 =) σβ(α) = α − β 2 Φ or hβ; αi = 1 =) β − α 2 Φ =) α − β 2 Φ. An application of this lemma is to justify the existence of unbroken \strings" of roots of the form β + nα; n 2 Z; q ≤ n ≤ r. Write this in more formally. 2.11. Definition. [Hum72, p 47] Given ∆ ⊆ Φ, we say that ∆ is a simple system or base if • ∆ is a basis for span Φ P • Every β 2 Φ can be written as β = kαα with α 2 ∆ and all kα 2 Z≥0 or all kα 2 Z≤0. We note that ∆ need not be unique and so whenever any definition de- pends on ∆, there is a choice being made that must be accounted for. 2.12. Definition. The height of β 2 Φ relative to ∆ is given by X ht(β) = kα α2∆ P where β = α2∆ kαα. Using this notion of height, we can define positive roots relative to ∆ as those with kα ≥ 0; 8α 2 ∆ and a similar notion for negative roots. Note that this partitions Φ = Φ+ t Φ−. We also note that we can define a partial order on the roots in Φ with a fixed ∆. 3 2.13. Definition. [Hum72, p 47] We say µ ≺ λ if and only if λ − µ is a sum of positive roots (equivalently, of simple roots) or λ = µ. We note also the following relationship between simple roots. 2.14. Lemma. [Hum72, p 47] If ∆ is a base of Φ, then (α; β) ≤ 0 for α 6= β in ∆, and α − β is not a root. Proof. For α; β 2 ∆, let (α; β) > 0. Then, α − β 2 Φ, which violates the second axiom of a base. In essence, this lemma gives us that the angle between two simple roots is obtuse. 2.15. Definition. Let us define the half sum of positive roots as 1 X ρ := β: 2 β0 This vector (not necessarily a root) has some important properties, such as the following 2.16. Proposition. [Hum72, p 50] σα(ρ) = ρ − α for all α 2 ∆. Now, we wish to show the following existence theorem. 2.17. Theorem. Every root system has a base. To do this, we will make use of the fact 2.18. Proposition. The union of finitely many hyperplanes cannot be all of n R for n ≥ 2. As a result of this proposition, we may choose a vector in our Euclidean space E which does not lie in any hyperplane perpendicular to a root. S 2.19. Definition. We call a vector z 2 E regular if z 2 E n α2Φ Pα, where Pα is the hyperplane orthogonal to the root α, and singular otherwise. 2.20. Definition. Given a vector z 2 E, we define Φ+(z) := fα 2 Φ j (z; α) > 0g + + We also say α 2 Φ (z) is decomposable if α = β1 + β2 for some βi 2 Φ (z). Otherwise, it is indecomposable. 2.21. Proposition. If z 2 E is regular, then Φ = Φ+(z) [ Φ+(−z) Thus, following [Hum72], we rewrite our theorem above in a more explicit manner. 2.22. Theorem. Let z 2 E be regular. Then, the set ∆(z) of all indecom- posable roots in Φ+(z) is a base of Φ and every base of Φ has the form ∆(z0) for some regular z0 2 E. 4 Proof. Let dim E ≥ 2, otherwise the theorem is immediate. Now,β 2 Φ =) β 2 Φ+(z) or −β 2 Φ+(z) by the proposition above, so we need only show X β = kαα; kα 2 Z≥0 α2∆+(z) Assume that is not the case. Then, we may pick a β 2 Φ+(z) such that + (z; β) is as small as possible. Since β 62 ∆(z), there is a β1; β2 2 Φ (z) such that β = β1 + β2 =) (z; β) = (z; β1) + (z; β2) > 0 but then either β1 or β2 is not a positive integral linear combination of + elements in ∆(z), contradicting βi 2 Φ (z). Now, we need only show the elements of ∆(z) are linearly independent. By the lemma 2.14,(α; β) ≤ 0 for α; β 2 ∆(z) with β 6= ±α. Thus, for cα 2 R, X X X cαα = 0 =) x := rαα = (−rβ)β α2∆(z) αjrα>0 βjrβ <0 X =) (x; x) = −rαrβ(α; β) ≤ 0 αjrα>0 βjrβ <0 =) x = 0 X =) 0 = (x; z) = rα · (α; z) αjrα>0 where (α; z) > 0 since α 2 Φ+(z) by definition, so it must be that all the rα = 0 and similarly for all rβ, thus giving linear independence. Thus, ∆(z) is a base of Φ = Φ+(z) [ Φ+(−z) since ∆(z) spans Φ+(z) and thus all of Φ, and it meets the base axioms by above.