¤ ¤

d

¤¤

d

¤

¤

„ „

dd

dd

d

d

„ „

Figure 3.11: A permutahedron for BC3 (Exercise 3.10.2).

Chapter 4

Classi¬cation

4.1 Generators and relations

Let W be a ¬nite re¬‚ection group and R = {r1 , . . . , rm } the set of simple

re¬‚ections in W . Denote mij = |ri rj |. Notice mii = 1 for all i.

Theorem 4.1.1 The group W is given by the following generators and

relations:

W = r1 , . . . , rm | (ri rj )mij = 1 .

Proof.

4.2 Decomposable re¬‚ection groups

Coxeter graph. By Theorem 4.1.1, a a ¬nite re¬‚ection group W is given

by the following generators and relations:

W = r1 , . . . , rm | (ri rj )mij = 1

where R = {r1 , . . . , rm } is the set of simple re¬‚ections in W and mij =

|ri rj |. Notice mii = 1 for all i.

Now we wish to associate with W and a system of simple re¬‚ections

R a graph G, called Coxeter graph, whose nodes are in one-to-one cor-

respondence with the simple re¬‚ections r1 , . . . , rn in R. If ri and rj are

two distinct re¬‚ections, then, if mij = |ri rj | > 2, the nodes ri and rj are

connected by an edge with mark mij on it. If mij = 2, that is, if ri and

rj commute, then there is no edge connecting ri and rj . We say that the

group W is indecomposable if the graph G is connected; otherwise W is

said to be decomposable.

83

84

$ $

$ $

rr rr

rr $$$$ rr $$$$

£ £

$$rr$' £ $$rr ' £

$ $

$

rr W

rr

$' W $'

r £ r £

$$ $ r

r $

e £1 £ e #1 £

c cE

d d

#

£e £ e £

d d

d

e£ d T£

©d

d

©

E ‚

e

T£ £

r3 d £ e d £ e

d e

d e

£e c£ e

D

c d d

d £ d £

r

d2 ' ©

E d £ E d £

d‚

d d

£ £

Removing a chamber D from a circular gallery. Here we use the relation

r3 r2 = r2 r3 r2 r3 r2 r3 which is a consequence of (r2 r3 )4 = 1.

$ $

$ $

r

rrr $$$$

rr $$$$ rr

rr £ £

$$$$r ' £ $$$$r ' £

r

rW r

rW

' '

r £ r £

$$ $

r $ r

e #1 £ e #1 £

cE

d d cE

e £ e £

dr d

2 d d

d T£ d T£

©d

‚ E ‚

e d e

£ £

d £ e d £ e

d e d e

c£ e c£ e

r2

d d

£

d £

'r3 d

©

E d £ d £

d d

£ £

Removing dead end and repeated chambers from a circular gallery. We use

2 2

the relations r2 = r3 = 1.

Figure 4.1: For the proof of Theorem 4.1.1.

Theorem 4.2.1 Assume that W is decomposable and let G1 , . . . , Gk be

connected components of G. Let Rj be the set of re¬‚ections corresponding

to nodes in the connected component Gj , j = 1, . . . , k. Let W j be the

parabolic subgroup generated by the set Rj . Then

W = W 1 — · · · — W k.

Proof. If i = j then any two re¬‚ections r ∈ Ri and r ∈ Rj commute,

hence

• the subgroups W i = Ri and W j = Rj commute elementwise.

By Theorem 3.8.1, the intersection of W j with the group generated by all

W i with i = j is the subgroup generated by Rj © i=j Ri = …, that is, the

identity subgroup:

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 85

• W j © W 1 , . . . , W j , . . . , W k = 1.

Finally, the subgroups W i generate W ,

• W = W 1, . . . , W k .

But these properties of the subgroups W i mean exactly that

W = W 1 — · · · — W k.

4.3 Classi¬cation of ¬nite re¬‚ection groups

4.4 Construction of root systems

In this section we shall construct, for each Coxeter graph of type

An , Bn , Cn , Dn , E6 , E7 , E8 , F4 , G2

a root system. An immediate computation shows that all these systems are

crystallographic. We do not consider the root systems of type H3 and H4 .

The interested reader may wish to consult the books [GB] and [H] which

contain a detailed discussion of these non-crystallographic root systems.

We notice only that the mirror system associated with the root system of

type H3 is the system of mirrors of symmetry of the regular icosahedron.

be the standard basis in Rn+1 .

Root system An . Let 1 , . . . , n+1

¦={ ’ | i, j = 1, . . . , n + 1, i = j },

i j

Π={ 2’ 1 , . . . , n+1 ’ n }.

¦ contains n(n + 1) vectors, all of whose are of equal length. Denote the

simple vectors as

ρ1 = ’ 1, ρ2 = ’ 2, . . . , ρn = ’

2 3 n+1 n

and take the root

ρ0 = n+1 ’ 1

= ρ1 + ρ2 + · · · + ρn .

The root ρ0 is called the highest root because it has, of all positive roots,

the longest expression in terms of the simple roots. The highest root plays

86

an exceptionally important role in many applications of the theory of root

systems, for example, in the representation theory of simple Lie algebras

and simple algebraic groups.

In the following diagram the black nodes form the Coxeter graph for

An ; an extra white node demonstrates the relations of the root ’ρ0 to the

simple roots. We use the following convention: if ± and β are two roots,

then their nodes are not connected if (±, β) = 0 (and the re¬‚ections s± and

sβ commute), and the nodes are connected by an edge if

(±, β) π

= ’ cos

|±||β| m

and m 3. In fact, m is the order of the product s± sβ and m 3 if and

only if the re¬‚ections s± and sβ do not commute. If m > 3 we write the

mark m on the edge.

’ρ0

c

s. . . s

s s

ρ1 ρ2 ρn’1 ρn

We know that the re¬‚ection group W for our root system is the sym-

metric group Symn+1 which acts by permuting the vectors i .

be the standard basis in Rn .

Root system Bn , n 2. Let 1, . . . , n

¦ = { ± i , ± i ± ej | i, j = 1, . . . , n, i < j },

Π = { 1 , 2 ’ 1 , . . . , n ’ n’1 }.

¦ contains 2n short roots ± i and 2n(n ’ 1) long roots ± i ± j , i < j.

It is convenient to enumerate the simple roots as

ρ1 = 1, ρ2 = ’ 1, . . . , ρn = ’ n’1 .

2 n

The highest root is

ρ0 = + n.

n’1

The extended Coxeter diagrams for the system of roots B2 and Bn with