and we shall freely use the relevant terminology. In particular, chambers

of Σ are open polyhedral cones”connected components of V H∈Σ H.

The closures of these cones are called closed chambers. Facets of chambers

(i.e. faces of maximal dimension) are panels and mirrors in Σ as walls.

Notice that every panel belongs to a unique wall. To fully appreciate this

architectural terminology (introduced by J. Tits), imagine a building built

out of walls of double-sided mirrors. Two chambers are adjacent if they

have a panel in common. Notice that every chamber is adjacent to itself.

Theorem 3.1.1 Every chamber C has the form ρ∈Π Vρ’ for some simple

system Π. Every panel of C belongs to one of the walls Hρ for a root ρ ∈ Π.

Vice versa, if ρ ∈ Π then Hρ © C is a panel of C.

Proof. Take any vector γ in the chamber C and consider the linear func-

tional f (») = ’(γ, »). Since γ does not belong to any mirror H± in Σ,

the functional f does not vanish on roots in ¦. Therefore the condition

f (±) > 0 determines a positive system ¦+ and the corresponding simple

system Π. Now consider the cone C de¬ned by the inequalities (», ρ) < 0

for all ρ ∈ Π. Obviously γ ∈ C and therefore C ⊆ C . If C = C

then some hyperplane H± , ± ∈ ¦, bounds C and intersects C nontriv-

ially. But ± = cρ ρ where all cρ are all non-negative or all non-positive,

and (γ, ±) = cρ (γ, ρ) cannot be equal 0. This contradiction shows that

C = C . The closure C of C is de¬ned by the inequalities (», ρ) 0 for

ρ ∈ Π which is equivalent to (», ±) 0 for ± ∈ “. Therefore the cone

63

64

Positive cone “

e

u e0

e

e

e e The fundamental chamber C is

e

e e de¬ned as the interior of the cone

e ee

'e e e e e e e e e E dual to the positive cone “, i.e.

e¨

as the set of vectors » such that

¨¨¨ e ¨

¨e ¨ (», γ) < 0 for all γ ∈ “.

¨¨ ¨ ¨

e¨

¨¨¨¨¨

¨¨e

¨¨

) …

¨

Fundamental

chamber C

Figure 3.1: The fundamental chamber.

$

$

r

rr $$ £

$

$

$$$ rr £ The Coxeter complex of type BC3 is

$

e £

d formed by all the mirrors of symmetry of

£

d e

the cube; here they are shown by their

£

d e

£e

d lines of intersection with the faces of the

£e

d cube.

d £

d

£

Figure 3.2: The Coxeter complex BC3 .

C is dual to the positive cone “ (see Figure 3.1) and every facet of C is

perpendicular to some edge of “, and vice versa. In particular, every panel

of C belongs to the wall Hρ for some simple root ρ ∈ Π.

The same argument works in the reverse direction: if Π is any simple

system then, since Π is linearly independent, we can ¬nd a vector γ such

that (γ, ρ) < 0 for all ρ ∈ Π. Then (γ, ±) = 0 for all roots ± ∈ ¦ and the

chamber C containing γ has the form C = ρ∈Π Vρ’ .

The set of all chambers associated with the root system ¦ is called the

Coxeter complex and will be denoted by C. See, for example, Figures 3.2

and 3.3. If Π is a simple system then the corresponding chamber is called

the fundamental chamber of C.

The following lemma is an immediate consequence of Lemma 1.2.3

Lemma 3.1.2 The union of two distinct adjacent closed chambers is con-

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 65

d

d

£ In the case that Σ is the system

d

£ d

of mirrors of symmetry of a regu-

£ 7

7

£ lar polytope ∆, the Coxeter complex

7

£

is basically the c subdivision of the

7

i

¢

e

i faces of ∆ by the mirrors of sym-

¢

e

ii metries of faces (here shown only on

2¢

e 222

d e2 ’ one face of the dodecahedron ∆).

3¢

’

d D 3

3

dD 3

Figure 3.3: Chambers and baricentric subdivision

1

The symmetry group of the tetra- „

„

hedron acts on its 4 vertices as

„

the symmetric group Sym4 . The

„

(23)

„––

(34)

re¬‚ections in the walls of the fun-

„ –– 4–

damental chamber are the trans- (12)

¡

„

positions (12), (23) and (34). ¡

„

„¡

Therefore they generate Sym4 .

„¡

3 2

Figure 3.4: Generation by simple re¬‚ections (Theorem 3.2.1).

vex.

3.2 Generation by simple re¬‚ections

Simple re¬‚ections. Let Π = { ρ1 , . . . , ρn } be a simple system of roots.

The corresponding re¬‚ections ri = sρi are called simple re¬‚ections.

Theorem 3.2.1 The group W is generated by simple re¬‚ections.

Proof. Set W = r1 , . . . , rn . We shall prove ¬rst that

the group W is transitive in its action on C.

Proof of the claim. The fundamental chamber C is bounded by panels lying

on the mirrors of the simple re¬‚ections r1 , . . . , rn . Therefore the neighbour-

ing chambers (i.e. the chambers sharing a common mirror with C) can be

obtained from C by re¬‚ections in these mirrors and equal r1 C, . . . , rn C.

66

Let now w ∈ W , then the panels of the chamber wC belong to the mirrors

of re¬‚ections wr1 w’1 , . . . , wrn w’1 . If D is a chamber adjacent to wC then

it can be obtained from wC by re¬‚ecting wC in the common mirror, hence

D = wri w’1 · wC = wri C for some i = 1, . . . , n. Notice that wri ∈ W . We

can proceed to move from a chamber to an adjacent one until we present

all chambers in C in the form wC for appropriate elements w ∈ W .

We can now complete the proof. If ± ∈ ¦ is any root and s± the

corresponding re¬‚ection then the wall H± bounds some chamber D. We

know that D = wC for some w ∈ W . The fundamental chamber C is

bounded by the walls Hρi for simple roots ρi (Theorem 3.1.1) and therefore

the wall H± equals wHρi for some simple root ρi . Thus s± = wri w’1 belongs

to W . Since the group W is generated by re¬‚ections s± we have W = W .

In the course of the proof we have obtained one more important result:

Corollary 3.2.2 The action of W on C is transitive.

This observation will be later incorporated into Theorem 3.5.1.

Exercises

3.2.1 Use Theorem 3.2.1 to prove the (well-known) fact that the symmetric

group Symn is generated by transpositions (12), (23), . . . , (n ’ 1, n) (see Fig-

ure 3.4).

3.2.2 Prove that the re¬‚ections

r1 = (12)(1— 2— ), . . . , rn’1 = (n ’ 1, n)(n ’ 1— , n— ), rn = (n, n— )

generate the hyperoctahedral group BCn .

3.3 Foldings

Given a non-zero vector ± ∈ V , the hyperplane H± = { γ ∈ V | (γ, ±) = 0 }

cuts V in two subspaces

+ ’

V± = { γ | (γ, ±) 0 } and V± = { γ | (γ, ±) 0}

intersecting along the common hyperplane H± . The folding in the direction

of ± is the map f± de¬ned by the formula

β if (β, ±) 0

f± (β) = .

s± β if (β, ±) < 0

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 67

f±

C

¡e

¡e

In the 2-dimensional case, a ¡ e

¡

R2 e

folding is exactly what its

¡ ¡

name suggests: the plane is ¡

e ±e

e 'e ¡

being folded on itself like a

e V+ e ¡ ’

V±

sheet of paper. e± e¡

H±

Figure 3.5: Folding

+ ’ +

Thus f± ¬xes all points in V± and maps V± onto V± symmetrically (see

Figure 3.5). Notice that f± is an idempotent map, i.e. f± f± = f± . The

folding f’± is called the opposite to f± . The re¬‚ection s± is made up of

two foldings f± and f’± :

s± = f± |V± ∪ f’± |V± .

+ ’

We say that a folding f covers a subset X ‚ V if X ⊆ f (V ).

By de¬nition, a folding of the chamber complex C is a folding along one

of its walls.