Exercises

2.5.1 Prove Lemma 2.5.2.

Hint: Find a regular n-gon such that W (¦) coincides with its symmetry group.

2.5.2 Prove that, in a root system in R2 , the lengths of roots can take at most

two values.

Hint: Use Exercise 2.3.6.

2.5.3 Describe planar root systems with 2 and 4 roots and the corresponding

re¬‚ection groups.

2.5.4 Use the observation that the root system G2 contains two subsystems of

type A2 to show that the dihedral group D12 contains two di¬erent subgroups

isomorphic to the dihedral group D6 .

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 49

2.5.5 Crystallographic root systems. For the root systems ¦ of types

A2 , B2 , C2 , G2 , sketch the sets Λ = Z¦ of points in R2 which are linear combi-

nations of roots in ¦ with integer coe¬cients,

a± ± | a± ∈ Z

Λ= .

±∈¦

Observe that Λ is a subgroup of R2 and, moreover, a discrete subgroup of R2 ,

that is, there is a real number d > 0 such that, for any » ∈ Λ, the circle

{ ± ∈ R2 | d(±, ») < d } contains no points from Λ other than ». We shall call

root systems in Rn with the analogous property crystallographic root systems.

2.6 Positive and simple systems

Positive systems. Let f : Rn ’’ R be a linear functional. Assume that

f does not vanish on roots in ¦, i.e. f (±) = 0 for all ± ∈ ¦. Then every

root ρ in ¦ is called positive or negative, according to whether f (ρ) > 0

or f (ρ) < 0. We shall write, abusing notation, ± > β if f (±) > f (β).

The system of all positive roots will be denoted ¦+ and called the positive

system. Correspondingly the negative system is denoted ¦’ . Obviously

¦ = ¦+ ¦’ .

Let “ denotes the convex polyhedral cone spanned by the positive sys-

tem ¦+ . We follow notation of Section 1.5 and call the positive roots s

directed along the edges of “ simple roots. The set of all simple roots

is called the simple system of roots and denoted Π; roots in Π are called

simple roots. It is intuitively evident that the cone “ is generated by sim-

ple roots, see also Lemma 1.5.3. In particular, every root φ in ¦+ can be

written as a non-negative combination of roots in Π:

φ = c1 ρ1 + · · · + cm ρm , ci 0, ρi ∈ Π.

Notice that the de¬nition of positive, negative, simple systems depends

on the choice of the linear functional f . We shall call a set of roots positive,

negative, simple, if it is so for some functional f .

Lemma 2.6.1 In a simple system Π, the angle between two distinct roots

is non-acute: (±, β) 0 for all ± = β in Π.

Proof. Let P be a two-dimensional plane spanned by ± and β. Denote

¦0 = ¦ © P . If γ, δ ∈ ¦0 then the re¬‚ection sγ maps δ to the vector

2(γ, δ)

sγ δ = δ ’ γ

(γ, γ)

50

Positive

cone “

$β

$$$

rr ± $$$$

d r$$$$$$

d $$$$$$ Planar root system ¦0 gener-

$$$$$ $

$

e $$$$

d$$$$$$

u

$ $$$$$ ated by two simple roots ± and

$$$$$$

e

$$$$$$

$$$$$$

$$$ $$

e$ $ β.

$$$$

$$$$$$

$$$$$

$$$$$ Plane P

$ e$$

$$$$$$

$$$$$$

e

$$$$$$

$$$$$$ spanned

…

e by

$

$$$$ $

$

$

$

©$

$

± and β

Figure 2.12: For the proof of Lemma 2.6.1

which obviously belongs to P and ¦0 . Hence every re¬‚ection sγ for γ ∈ ¦0

obviously maps P to P and ¦0 to ¦0 . This means that ¦0 is a root system

in P and ¦+ © P is a positive system in ¦0 .

Moreover, the convex polyhedral cone “0 spanned by ¦+ = ¦+ © P is

0

contained in “ © P . Since ± and β are obviously directed along the edges

of “ © P (see also Lemma 1.5.4) and belong to “0 , “0 = “ © P and ± and β

belong to a simple system in ¦0 , see Figure 2.12. Therefore the lemma is

reduced to the 2-dimensional case, where it is self-evident, see Figure 2.13.

Notice that our proof of Lemma 2.6.1 is a manifestation of a general

principle: surprisingly many considerations in roots systems can be reduced

to computations with pairs of roots.

Theorem 2.6.2 Every simple system Π is linearly independent. In par-

ticular, every root β in ¦ can be written, and in a unique way, in the form

c± ± where ± ∈ Π and all coe¬cients c± are either non-negative (when

β ∈ ¦+ ) or non-positive (when β ∈ ¦’ ).

Proof. Assume, by way of contradiction, that Π is linearly dependent

and

a± ± = 0

±∈Π

where some coe¬cient a± = 0. Rewrite this equality as bβ β = cγ γ

where the coe¬cients are strictly positive and the sums are taken over

disjoint subsets of Π. Set σ = bβ β. Since all roots β are positive, σ = 0.

But

0 (σ, σ) = bβ β, cγ γ = bβ cγ (β, γ) 0,

γ γ

β β

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 51

positive

positive

cone “

halfplane, f (γ) > 0

T β

s

d

d

¦+

d

d $

$

$$$$

d

d $

$ $$

±' E

$$ d

$$

$$$ d

d negative

d halfplane, f (γ) < 0

d

©’

d

‚

¦

c

Figure 2.13: For the proof of Lemma 2.6.1. In the 2-dimensional case the

obtuseness of the simple system is obvious: the roots ± and β are directed along

the edges of the convex cone spanned by ¦+ and the angle between ± and β is

at least π/2.

because all individual scalar products (β, γ) are non-positive by Lemma 2.6.1.

Therefore σ = 0, a contradiction.

Corollary 2.6.3 All simple systems in ¦ contain an equal number of

roots.

Proof. Indeed, it follows from Theorem 2.6.2 that a simple system is a

maximal linearly independent subset of ¦.

The number of roots in a simple system of the root system ¦ is called

the rank of ¦ and denoted rk ¦. The subscript n in the standard notation

for root systems An , Bn , etc. (which will be introduced later) refers to their

ranks.

Exercises

2.6.1 Prove that, in a planar root system ¦ ‚ R2 , all positive (correspondingly,

simple) systems are conjugate under the action of the re¬‚ection group W =

W (¦).

2.7 Root system An’1.

Permutation representation of Symn . Let V be the real vector space

Rn with the standard orthonormal basis 1 , . . . , n and the corresponding

coordinates x1 , . . . , xn .

52

The group W = Symn acts on V in the natural way, by permuting the

n vectors 1 , . . . , n :

w i = wi ,

which, obviously, induces an action of W on ¦. The action of the group

W = Symn on V = Rn preserves the standard scalar product associated

with the orthonormal basis 1 , . . . , n . Therefore W acts on V by orthogonal

transformations.

In its action on V the transposition r = (ij) acts as the re¬‚ection in

the mirror of symmetry given by the equation xi = xj .

Lemma 2.7.1 Every re¬‚ection in W is a transposition.

Proof. The cycle (i1 · · · ik ) has exactly one eigenvalue 1 when restricted

to the subspace R i1 • · · · • R ik , with the eigenvector i1 + · · · + ik . It

follows from this observation that the multiplicity of eigenvalue 1 of the

permutation w ∈ Symn equals the number of cycles in the cycle decom-

position of w (we have to count also the trivial one-element cycles of the

form (i)). If w is a re¬‚ection, then the number of cycles is n ’ 1, hence w

is a transposition.

Regular simplices. The convex hull ∆ of the points 1, . . . , n is the

convex polytope de¬ned by the equation and inequalities

x1 + · · · + xn = 1, x1 0, . . . , xn 0.

Since the group W = Symn permutes the vertices of ∆, it acts as a group

of symmetries of ∆, W Sym ∆. We wish to prove that actually W =

Sym ∆. Indeed, any symmetry s of ∆ acts on the set of vertices as some

permutation w ∈ Symn , hence the symmetry s’1 w ¬xes all the vertices

1 , . . . , n of ∆ and therefore is the identity symmetry.

The polytope ∆ is called the regular (n ’ 1)-simplex. When n = 3, ∆ is

an equilateral triangle lying in the plane x1 + x2 + x3 = 1 (see Figure 2.14),

and when n = 4, ∆ is a regular tetrahedron lying in the 3-dimensional

a¬ne Euclidean space x1 + x2 + x3 + x4 = 1.

The root system An’1 . We shall introduce the root system ¦ of type

An’1 , as the system of vectors in V = Rn of the form i ’ j , where

i, j = 1, 2, . . . , n and i = j. Notice that ¦ is invariant under the action of

W = Symn on V .

In its action on V the transposition r = (ij) acts as the re¬‚ection in

the mirror of symmetry perpendicular to the root ρ = i ’ j . Hence ¦ is

a root system. Since the symmetric group is generated by transpositions,

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 53

x

T3

(0, 0, 1)

The transposition (12) acts on

¢dh

¢h d

R3 as the re¬‚ection in the mirror

¢h d

x1 = x2 and as a symmetry of

¢h d

¢ the equilateral triangle with the