· rl’1

Hint: use induction on l.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 73

$$

r r2rr

$r2 $$r1 £

r3

$

rr $$ £

r3

$$

$$ rr r3 r £

$$ r2 r3 r2 r

$ 1

$ £

$ r

£

r1 r1

d e r3

£r2

d e

re £

d

r3

£

2

r2 r2

d e r3

r1

£

d r3 e

r2£

d £ e

r3 r3

d e

r2

£ r3 e

d

e

r1 £

r2 dr2

£

d

r3

£

d

£

d

d £

Figure 3.7: Labelling of panels in the Coxeter complex BC3 .

3.6 Labelling of the Coxeter complex

We shall use the simple transitivity of the action of the re¬‚ection group

W on its Coxeter complex C to label each panel of the Coxeter complex C

with one of the simple re¬‚ections r1 , . . . , rn ; the procedure for labelling is

as follows.

First we label the panels of the fundamental chamber C by the corre-

sponding simple re¬‚ections. If D is a chamber in C, then there is unique

element w ∈ W which sends C to D = wC. If Q is a panel of D, we assign

to the panel Q of D the same label as that of the panel P = w’1 Q of C.

However, we need to take care of consistency of labelling: the panel Q

belongs to two adjacent chambers D and D . If we label the panels of D

by the same rule, will the label assigned to Q be the same? Let r be the

simple re¬‚ection in the panel P and C = rC the chamber adjacent to C

and sharing the panel P with C. Since the action of W on C preserves

adjacency of chambers, D = wC = wrC. Hence wr is a unique element

of W which send C to D , and we assign to the panel Q the label of the

panel (wr)’1 Q of C. But rP = P , hence (wr)’1 Qrw’1 Q = rP = P , and

Q gets the same label as before.

If a common panel of two chambers D and E is labelled ri , we shall

say that D and E are ri -adjacent. This includes the case D = E, so that

every chamber is ri -adjacent to itself.

The following observation is immediate.

Proposition 3.6.1 The action of W preserves the labelling of panels in

74

the Coxeter complex C.

Moreover, we can now start to develope a vocabulary for translation of

the geometric properties of the Coxeter complex C into the language of the

corresponding re¬‚ection group W .

Theorem 3.6.2 Let C be a fundamental chamber in the Coxeter complex

C of a re¬‚ection group W . The map

w ’ wC

is a one-to-one correspondence between the elements in W and chambers

in C. Two distinct chambers C and C are ri -adjacent if and only if the

corresponding elements w and w are related as w = wri .

Now the description of canonical galleries given in Theorems 3.4.1 and

3.4.2 can be put in a much more convenient form.

Let “ = { C0 , . . . , Cl } be a gallery and let rik the label of the common

panel of the consequent chambers Ck’1 and Ck , k = 1, . . . , l. Then we say

that “ has type ri1 , . . . , ril .

Theorem 3.6.3 Let “ = { C0 , . . . , Cl } be a gallery of type ri1 , . . . , ril con-

necting the fundamental chamber C = C0 and a chamber D = Cl . Set

rik if Ck’1 = Ck

rik =

ˆ

1 if Ck’1 = Ck .

Then

D = ri1 · · · ril C.

ˆ ˆ

For all k = 1, . . . , l, the element of W corresponding to the chamber Ck is

ri1 · · · rik . In particular, if the galery “ does not stutter, then we have, for

ˆ ˆ

all k, rik = rik and “ is a canonical gallery for the word w = ri1 · · · ril .

ˆ

Proof. The proof is obvious.

3.7 Isotropy groups

We remain in the standard setting of our study: ¦ is a root system in Rn ,

Σ is the corresponding mirror system and W is the re¬‚ection group.

If ± is a vector in Rn , its isotropy group or stabiliser , or centraliser (all

these terms are used in the literature) CW (±) is the group

CW (±) = { w ∈ W | w± = ± };

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 75

if X ⊆ Rn is a set of vectors, then its isotropy group or pointwise centraliser

in W is the group

CW (X) = { w ∈ W | w± = ± for all ± ∈ X }.

Theorem 3.7.1 In this notation,

(1) The isotropy group CW (X) of a set X ‚ Rn is generated by those re-

¬‚ections in W which it contains. In other words, CW (X) is generated

by re¬‚ections sH , H ∈ Σ, whose mirrors contain the set X.

(2) If X belongs to the closure C of the fundamental chamber then the

isotropy group CW (X) is generated by the simple re¬‚ections it con-

tains.

Proof. Consider ¬rst the case when X = { ± } consists of one vector.

Write W = CW (±). If the vector ± does not belong to any mirror in Σ

then it lies in one of the open chambers in C, say D, and wD = D for

any w ∈ W . It follows from the simple transitivity of W on the Coxeter

complex C (Theorem 3.5.1) that W = 1, and the theorem is true since W

contains no re¬‚ections.

Now denote by Σ the set of all mirrors in Σ which contain ±. Obviously,

Σ is a closed mirror system and is invariant under the action of W .

Consider the set C of all chambers D such that ± ∈ D. Notice that C

is invariant under the action of W .

If D ∈ C and P a panel of D containing ± then the wall H of P belongs

to Σ , and the chamber D adjacent to D via the panel P belongs to C .

Also, if two chambers D, D ∈ C are adjacent in C and have the panel P

in common, then P = D © D contains ±, and the wall H containing the

panel P and its closure P belongs to Σ .

These observations allow us to prove that any two chambers D and D

in C can be connected by a gallery which belongs to C . Indeed, let

D = D0 , D1 , . . . , Dl = D

be a geodesic gallery connecting D and D . If one of the chambers in the

gallery, say Dk , does not belong to C , then select the minimal k with this

property and look at the wall H separating Dk’1 and Dk . The chambers

D, Dk’1 , D lie on the same side of the wall H as the point ±. But a

geodesic gallery intersects each wall only once. Hence the entire gallery

belongs to C .

Now take an arbitrary w ∈ W and consider a gallery D0 , . . . , Dl in C

connecting the chambers D = D0 and Dl = wD. If si is the re¬‚ection in the

76

common panel of the consecutive chambers Di’1 and Di , i = 1, . . . , l, then

D = sl · · · s1 D. Since W acts on C simply transitively, w = sl · · · s1 . But,

for each i, si ∈ W , therefore the group W is generated by the re¬‚ections

it contains. This proves (1) in our special case. The statement (2) for

X = {±} follows from the observation that if D = C is the fundamental

chamber then the proof of the Theorem 3.2.1 can be repeated word for

word for W and C and shows that W is generated by re¬‚ections in the

walls of the fundamental chamber C, i.e. by simple re¬‚ections.

Now consider the general case. If every point in X belongs to every

mirror in Σ then CW (X) = W and the theorem is trivially true. Otherwise

take any ± in X such that the system Σ of mirrors containing ± is strictly

smaller than Σ. Then CW (X) CW (±) and W = CW (±) is itself the

re¬‚ection group of Σ . We can use induction on the number of mirrors in

Σ, and application of the inductive assumption to Σ completes the proof.

Exercises

3.7.1 For the symmetry group of the cube ∆ = [’1, 1]3 , ¬nd the isotropy

groups

(a) of a vertex of the cube,

(b) of the midpoint of an edge,

(c) of the center of a 2-dimensional face.

3.7.2 Let ¦ be the root system of the ¬nite re¬‚ection group W and ± ∈ ¦.

Prove that the isotropy group CW (±) is generated by the re¬‚ections sβ for all

roots β ∈ ¦ orthogonal to ±.

3.7.3 The centraliser CW (u) of an element w ∈ W is the set of all elements in

W which commute with u:

CW (w) = { v ∈ W | vu = uv }.

Let s± be the re¬‚ection corresponding to the root ± ∈ ¦. Prove that

CW (s± ) = s± — sβ | β ∈ ¦ and β orthogonal to ± .

3.7.4 Let W = Symn and r = (12). Prove that

CW (r) = (12) — (34), (45), . . . , (n ’ 1, n)

and is isomorhic to Sym2 — Symn’2 .

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 77

3.7.5 Let ∆ be a convex polytope and assume that its group of symmetries

contains a subgroup W generated by re¬‚ections. If “ is a face of ∆, prove that

the set-wise stabiliser of “ in W

StabW (“) = { w ∈ W | w“ = “ }

is generated by re¬‚ections.

3.8 Parabolic subgroups

Let Π be a simple system in the root system ¦ and r1 , . . . , rm the cor-

responding system of simple re¬‚ections. Denote I = { 1, . . . , m, }. For a

subset J ⊆ I denote

WJ = ri | i ∈ J ;

subgroups WJ are called standard parabolic subgroups of W . Notice WI =

W and W… = 1.

For each i = 1, . . . , m, denote by P i the (closed) panel of the closed

fundamental chamber C corresponding to the re¬‚ection ri , and set

PJ = PJ .

i∈J

By virtue of Theorem 3.7.1,

WJ = CW (P J ).

We are now in a position to obtain a very easy proof of the following

beautiful properties of parabolic subgroups.

Theorem 3.8.1 If J and K are subsets of I then

WJ∪K = WJ , WK

and

WJ©K = WJ © WK .

Proof. The ¬rst equality is obvious, the second one follows from the

observation that

WJ © WK = CW (P J ) © CW (P K ) = CW (P J ∪ P K )