Proposition 3.3.1 A folding f of C sends chambers to chambers and pre-

serves adjacency: if C and D are two adjacent chambers then their images

f (C) and f (D) are also adjacent. (Remember that, by de¬nition of adja-

cency, this includes the possibility that f (C) = f (D).)

Exercises

3.3.1 When you fold a sheet of paper, why is the line along which it is folded

straight?

3.3.2 There are three foldings of the chamber complex BC2 such that their

composition maps the chamber complex onto one of its chambers. What is the

minimal number of foldings needed for folding the chamber complex BC3 onto

one chamber?

3.4 Galleries and paths

Galleries. Given two chambers C and D, we can always ¬nd a sequence

G of chambers C = C0 , C1 , . . . , Cl’1 , Cl such that every two consequtive

68

chambers Ci’1 and Ci are adjacent. We shall call G a gallery connecting

the chambers C and D. Notice that our de¬nition of adjacency allows that

two adjacent chambers coincide. This means that we also allow repetition

of chambers in a gallery: it could happen that Ci’1 = Ci . We shall say in

this situation that the gallery stutters at chamber Ci . The number l will

be called the length of the gallery G.

Notice that if si is the re¬‚ection in a common wall of two adjacent

chambers Ci’1 and Ci then either Ci = si Ci’1 or Ci = Ci’1 .

Given w ∈ W , we wish to describe a canonical way of connecting the

fundamental chamber C and the chamber D = wC by a gallery. We

know that W is generated by the fundamental re¬‚ections r1 , . . . , rn , i.e.

the re¬‚ections in the walls of the fundamental chamber C. The minimal

number l such that w is the product of some l fundamental re¬‚ections is

called the length of w and denoted as l(w).

Let w = ri1 · · · ril . We leave to the reader to check the following group

theoretical identity: since all ri are involutions,

ri ···ri1 ri ···ri1 r i

ri1 · · · ril = ril l’1 l’2

· ril’1 · · · ri21 · ri1 .

r ···ri1

Denote sj = rijj’1 , then w = sl · · · s1 and, moreover,

sj · · · s1 = ri1 · · · rij for j = 1, . . . , l.

De¬ne by induction C0 = C and, for i = 1, . . . , l, Cj = sj Ci’1 , so that

Cj = sj · · · s1 C0

= ri1 · · · rij C0 for j > 0,

Cl = ri1 · · · ril C0

= wC = D.

Notice that s1 = r1 is the re¬‚ection in the common wall of the chambers

C0 and C1 . Next, sj for j > 1 is written as

ri ···ri

sj = rij j’1 1

= (ri1 · · · rij’1 )rij (ri1 · · · rij’1 )’1 .

By Lemma 2.1.3, since rij is a re¬‚ection in a panel, say H, of the funda-

mental chamber C = C0 , sj is the re¬‚ection in the panel ri1 · · · rij’1 H of

the chamber ri1 · · · rij’1 C0 = Cj’1 . Since sj Cj’1 = Cj ,

sj is the re¬‚ection in the common panel of the chambers Cj’1

and Cj .

Summarising this procedure we obtain the following result; it will show

us the right way in the labyrinth of mirrors.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 69

Theorem 3.4.1 Let w = ri1 · · · ril be an expression of w ∈ W in terms of

simple re¬‚ections ri . Let C be the fundamental chamber and D a chambers

in C such that D = wC. Then there exists a unique gallery C0 , C1 , . . . , Cl

connecting C = C0 and D = Cl with the following property:

r ···r

sj = rijj’1 i1 is the re¬‚ection in the common wall of Cj’1 and

Cj , j = 1, . . . , l and w = sl · · · s1 .

The gallery C0 , . . . , Cl constructed in Theorem 3.4.1 will be called the

canonical w-gallery starting at C = C0 .

We can reverse the above arguments and obtain also the following re-

sult.

Theorem 3.4.2 Let C0 , . . . , Cl be a gallery connecting the fundamental

chamber C = C0 and a chamber D = Cl . Assume that the gallery does

not stutter at any chamber, that is, no two consequent chambers Ci and

Ci+1 coincide. Let si be the re¬‚ection in the common wall of Ci’1 and Ci ,

i = 1, . . . , l

Then D = wC for w = sl · · · s1 ,

Cj = sj · · · s1 C0 for all j = 1, . . . , l,

and there exists an expression w = ri1 · · · ril of w in terms of simple re¬‚ec-

tions ri such that, for all j = 1, . . . , l,

r ···ri1

sj = rijj’1 .

3.5 Action of W on C

In this section we shall prove arguably the most important property of the

Coxeter complex.

Theorem 3.5.1 The group W is simply transitive on C, i.e. for any two

chambers C and D in C there exists a unique element w ∈ W such that

D = wC.

Paths. We shall call a sequence of points γ0 , . . . , γl a path if

• the consecutive points γi’1 and γi are contained in adjacent chambers

Ci’1 and Ci ;

70

• if Ci’1 = Ci then γi’1 = γi ;

• if Ci’1 = Ci and si is the re¬‚ection in the common panel of Ci’1 and

Ci then γi = si γi’1 .

The number l is called the length of the path. Set w = sl · · · s0 . Since

γl = sl · · · s1 γ0 = wγ0 ,

the sequence of chambers C0 , C1 , . . . , Cl is the canonical w-gallery, and, by

Theorem 3.4.2, w can be expressed as a product of l simple re¬‚ections. So

we have the following useful lemma.

Lemma 3.5.2 Given a path γ0 , γ1 , . . . , γl , there exists w ∈ W such that

γl = wγ0 and l(w) l.

Notice the important property of paths: since we know that the union

of two distinct adjacent closed chambers is convex (Lemma 3.1.2), the

wall Hsi is the only wall intersecting the segment [γi’1 γi ]. Therefore the

following lemma holds.

Lemma 3.5.3 If γ0 , . . . , γl is a path connecting the points γ0 and γl lying

on the opposite sides of the wall H. Then the path intersects H in the sense

that, for some two consecutive points γi’1 and γi , the wall H intersects

the segment [γi’1 , γi ] and

• the common panel of the chambers Ci’1 and Ci containing γi’1 and

γi , respectively, belongs to H;

• γi’1 and γi are symmetric in H.

Paths and foldings. As often happens in the theory of re¬‚ection groups,

an important technical result we wish to state now can be best justi¬ed by

referring to a picture (Figure 3.6).

Lemma 3.5.4 Assume that the starting point ± = γ0 and the end point

ω = γl of a path γ0 , . . . , γl lie on one side of a wall H. If the wall H

intersects the path, that is, one of the points γi lies on the opposite side of

H from ±, then the path can be replaced by a shorter path with the same

starting and end points, and such that it does not intersect the wall H.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 71

H H

$ $

$ $

r

rrr $$$$ rr $$$$

rr £ rr £

$$$r$' £ £

$$$r

$

rr

rr

' W '

$$$ $$ r £

r £

r $ r

e £ £ £

c c

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± £e £ £

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d

d d ± e

r e£ e£

d

© d

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d ω £e d ω T £e

T £

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Figure 3.6: For the proof of Lemma 3.5.4: the folding in a wall which intersects

a path converts the path to a shorter one.

Proof. See the quite self-explanatory Figure 3.6. A rigorous proof fol-

lows. However, it can be skipped in the ¬rst reading.

Assume that the path intersects the wall H at the segment

[γp1 ’1 γp1 ]. Then, in view of Lemma 3.5.3, the path should intersect

the wall at least once more, say at the segments

[γp2 ’1 γp2 ], . . . , [γpk ’1 γpk ].

Let C0 , . . . , Cl be the gallery corresponding to our path, so that

γi ∈ Ci . Take the folding f in H onto the half space containing ±

and β and consider the path f (γ0 ), f (γ1 ), . . . , f (γl ) and the gallery

f (C0 ), f (C1 ), . . . , f (Cl ). In this new gallery and new path we have

repeated chambers, namely.

f (Cp1 ’1 ) = f (Cp1 ), . . . , f (Cpk ’1 ) = f (Cpk )

and points

f (γp1 ’1 ) = f (γp1 ), . . . , f (γpk ’1 ) = f (γpk )

After deleting the duplicate chambers and points and changing the

numeration we obtain a shorter gallery C0 , C1 , . . . , Cm and a path

γ0 , γ1 , . . . , γm such that γ0 = γ0 , γm = γl and for all i = 1, . . . , m,

• γi ∈ Ci ;

• Ci’1 and Ci are adjacent;

• if si is the re¬‚ection in the common wall of Ci’1 and Ci then

γi = si γi’1 .

But then = γ0 , . . . , γm is a shorter path connecting ± and ω.

72

Simple transitivity of W on C: Proof of Theorem 3.5.1. In view

of Corollary 3.2.2 we need to prove only the uniqueness of w. If D =

w1 C and D = w2 C for two elements w1 , w2 ∈ W and w1 = w2 , then

’1 ’1

w2 w1 C = C. Denote w = w2 w1 ; we wish to prove w = 1. Assume, by

way of contradiction, that w = 1. Of all expressions of w in terms of the

generators r1 , . . . , rn we take a shortest, w = ri1 · · · ril , where l = l(w) is

the length of w. Since w = 1, l = 0. Now of all w ∈ W with the property

that wC = C choose the one with smallest length l.

We can assume without loss of generality that C is the fundamental

chamber. Let now C0 , C1 , . . . , Cl be the canonical w-gallery connecting C

with C.

The vectors from the open cone C obviously span the vector space V ,

so the non-trivial linear transformation w cannot ¬x them all. Take γ ∈ C

such that wγ = γ and consider the sequence of points γi , i = 0, 1, . . . , l

de¬ned by γ0 = γ and γi = si γi+1 for i > 0. Then γi ∈ Ci . The sequence

γ0 , γ1 , . . . , γl is a path and links the end points γ0 = γ and γl = wγ. Now

consider the wall H = Hs1 . Since γ0 and γl both lie in C, they lie on the

same side of H. But the point γ1 = s1 γ0 lies on the opposite side of H

from γ. Hence, by Lemma 3.5.4, there is a shorter path connecting γ and

wγ and, by Lemma 3.5.2, an element w ∈ W with w ± = ω and smaller

length l(w ) < l than that of w. This contradiction completes the proof of

the theorem.

Since we have a one-to-one correspondence between positive systems,

simple systems and fundamental chambers, we arrive at the following re-

sult.

Theorem 3.5.5 The group W acts simply transitively on the set of all

positive (simple) systems in ¦.

Another important result is the following observation: for every root

± ∈ ¦ the mirror H± bounds one of the chambers in in C. Since every

chamber corresponds to some simple system in ¦ and all simple systems

are conjugate by Theorem 3.5.5, we come to

Theorem 3.5.6 Let ¦ be a root system, Π a simple system in ¦ and W

the re¬‚ection group of ¦. Every root ± ∈ ¦ is conjugate, under the action

of W , to a root in Π.

Exercises

3.5.1 Prove, for involutions r1 , . . . , rl in a group G, the identity

r ···r1 r ···r1 r

r1 · · · rl = rl l’1 l’2