e2kπi/n = cos 2kπ/n + i sin 2kπ/n, k = 0, 1, . . . , n ’ 1

in the complex plane C are vertices of a regular n-gon ∆. Prove that the maps

r : z ’ z · e2πi/n ,

t : z ’ z,

¯

where ¯ denotes the complex conjugation, generate the group of symmetries of

∆.

2.3.8 Use the idea of the proof of Theorem ?? to ¬nd the orders of the groups

of symmetries of the regular tetrahedron, cube, dodecahedron.

5

We use ˜left™ notation for action, so when we apply the composition st of two

transformations s and t to a point, we apply t ¬rst and then s: (st)A = s(tA).

44

2.4 Root systems

Mirrors and their normal vectors. Consider a re¬‚ection s with the

mirror H. If we choose the orthogonal system of coordinates in V with the

origin O belonging to H then s ¬xes O and thus can be treated as a linear

orthogonal transformation of V . Let us take a nonzero vector ± perpen-

dicular to H then, obviously, R± = H ⊥ is the orthogonal complement of

H in V , s preserves H ⊥ and therefore sends ± to ’±. Then we can easily

check that s can be written in the form

2(β, ±)

s± β = β ’ ±,

(±, ±)

where (±, β) denotes the scalar product of ± and β. Indeed, a direct com-

putation shows that the formula holds when β ∈ H and when β = ±. By

the obvious linearity of the right side of the formula with respect to β, it

is also true for all β ∈ H + R± = V .

Also we can check by a direct computation (left to the reader as an

exercise) that, given the nonzero vector ±, the linear transformation s± is

orthogonal, i.e. (s± β, s± γ) = (β, γ) for all vectors β and γ. Finally, s± = sc±

for any nonzero scalar c.

Notice that re¬‚ections can be characterized as linear orthogonal trans-

formations of Rn with one eigenvalue ’1 and (n ’ 1) eigenvalues 1; the

vector ± in this case is an eigenvector corresponding to the eigenvalue ’1.

Thus we have a one-to-one correspondence between the three classes of

objects:

• hyperplanes (i.e. vector subspaces of codimension 1) in the Euclidean

vector space V ;

• nonzero vectors de¬ned up to multiplication by a nonzero scalar;

• re¬‚ections in the group of orthogonal transformations of V .

The mirror H of the re¬‚ection s± will be denoted by H± . Notice that

H± = Hc± for any non-zero scalar c.

Notice, ¬nally, that orthogonal linear transformations of the Euclidean

vector space V (with the origin O ¬xed) preserve the relations between

mirrors, vectors and re¬‚ections.

Root systems. Traditionally closed systems of re¬‚ections were studied

in the disguise of root systems. By de¬nition, a ¬nite set ¦ of vectors in V

is called a root system if it satis¬es the following two conditions:

(1) ¦ © Rρ = { ρ, ’ρ } for all ρ ∈ ¦;

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 45

¦ ¦

s

d T

T

d s

d

d d

' d E ' d E

d d

d d

©

‚

d

cd

© d

‚ c

Figure 2.8: If ¦ is a root system then the vectors ρ/|ρ| with ρ ∈ ¦ form the

root system ¦ with the same re¬‚ection group. We are not much interested in

lengths of roots and in most cases can assume that all roots have length 1.

(2) sρ ¦ = ¦ for all ρ ∈ ¦.

The following lemma is an immediate corollary of Lemma 2.1.3.

Lemma 2.4.1 Let Σ be a ¬nite closed system of mirrors. For every mirror

H in Σ take two vectors ±ρ of length 1 perpendicular to H. Then the

collection ¦ of all these vectors is a root system. Vice versa, if ¦ is a root

system then { Hρ | ρ ∈ ¦ } is a system of mirrors.

Proof. We need only to recall that a re¬‚ection s, being an orthogonal

transformation, preserves orthogonality of vectors and hyperplanes: if ρ is

a vector and H is a hyperplane then ρ ⊥ H if and only if sρ ⊥ sH.

Also we can restate Lemma 2.2.1 in terms of root systems.

Lemma 2.4.2 Let ¦ be a root system. Then the group W generated by

re¬‚ections sρ for ρ ∈ ¦ is ¬nite.

Exercises

2.4.1 Prove, by direct computation, that the linear transformation s± given

by the formula

2(β, ±)

s± β = β ’ ±,

(±, ±)

is orthogonal, that is,

(s± β, s± β) = (β, β)

for all β ∈ V .

46

2.4.2 Let ¦ be a root system in the Euclidean space V and U < V a vector

subspace of V . Prove that ¦ © U is a (possibly empty) root system in U .

2.4.3 Let V1 and V2 be two subspaces orthogonal to each other in the real

Euclidean vector space V and ¦i be a root system in Vi , i = 1, 2. Prove that

¦ = ¦1 ∪ ¦2 is a root system in V1 • V2 ; it is called the direct sum of ¦1 and ¦2

and denoted

¦ = ¦ 1 • ¦2 .

2.4.4 We say that a group W of orthogonal transformations of V is essential

if it acts on V without nonzero ¬xed points. Let ¦ be a root system in V , ¦

and W the corresponding system of mirrors and re¬‚ection groups. Prove that

the following conditions are equivalent.

• ¦ spans V .

• The intersection of all mirrors in Σ consists of one point.

• W is essential on V .

2.5 Planar root systems

We wish to begin the development of the theory of root systems with

referring to the reader™s geometric intuition.

Lemma 2.5.1 If ¦ is a root system in R2 then the angles formed by pairs

of neighbouring roots are all equal. (See Figure 2.9.)

Proof of this simple result becomes self-evident if we consider, instead of

roots, the corresponding system Σ of mirrors, see Figure 2.10. The mirrors

in Σ cut the plane into corners (later we shall call them chambers), and

adjacent corners, with the angles φ and ψ, are congruent because they

are mirror images of each other. Therefore all corners are congruent. But

the angle between neighbouring mirrors is exactly the angle between the

corresponding roots.

Lemma 2.5.2 If a planar root system ¦ contains 2n vectors, n ≥ 1, then

the re¬‚ection group W (¦) is the dihedral group D2n of order 2n.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 47

T

The fundamental property of planar

s

d

d

d root systems: the angles ψ formed

d by pairs of neighbouring roots are

ψ

d

all equal. If the root system con-

' d E

d tains 2n vectors then ψ = π/n and

d

the re¬‚ection group is the dihedral

d

group D2n of order 2n.

d

d

©

d

‚

c

Figure 2.9: A planar root system (Lemma 2.5.1).

The fact that the angles formed by

d

d

d pairs of neighbouring roots are all

d equal becomes obvious if we con-

±

d

sider the corresponding system of

d

d β mirrors: ± = β because the ad-

d

jacent angles are mirror images of

d

each other.

d

d

d

Figure 2.10: A planar mirror system (for the proof of Lemma 2.5.1).

48

Proof left to the reader as an exercise.

We see that a planar root system consisting of 2n vectors of equal length

is uniquely de¬ned, up to elation of R2 . We shall denote it I2 (n). Later we

shall introduced planar root systems A2 (which coincides with I2 (3)) as a

part of series of n-dimensional root systems An . In many applications of

the theory of re¬‚ection groups the lengths of roots are of importance; in

particular, the root system I2 (4) associated with the system of mirrors of

symmetry of the square, comes in two versions, named B2 and C2 , which

contain 8 roots of two di¬erent lengths, see Figure 2.17 in Section 2.8.

Finally, the regular hexagon gives rise to the root system of type G2 , see

Figure 2.11.

T

t

t

t

r ¨

B

“

trr 0 t ¨¨

t r r ¨¨

t

t' r¨ E

t

t ¨¨rr t

¨ rr

¨t t

¨ rt

%

¨ r

j

t) ”

t

t

t

c