Exercises

For every of the root systems An , Bn , Cn , Dn , E6 , E7 , E8 , F4 , G2 :

4.4.1 Check the crystallographic condition.

4.4.2 Find the decomposition of the highest root with respect to the simple

roots.

4.4.3 Check, by a direct computation, that the extended Coxeter diagrams are

drawn correctly.

4.4.4 The sets ¦long and ¦short of all long (correspondingly, short) roots in a

root system ¦ are root systems on their own. Identify their types when ¦ is ot

type Bn , Cn or F4 .

4.5 Orders of re¬‚ection groups

In this section we shall use information about the root systems accumulated

in Section 4.4 to determine the orders of the ¬nite re¬‚ection groups

An , BCn , Dn , E6 , E7 , E8 F4 G2 .

Notice that the group A1 , obviously, has order 2. The groups A2 , BC2 ,

G2 are the dihedral groups of orders 6, 8, 12, correspondingly.

Let ¦ be one of the root systems listed above and W its re¬‚ection

group. To work out the order of W we need ¬rst to study the action of W

on ¦.

Lemma 4.5.1 The long (respectively, short) roots in ¦ are conjugate un-

der the action of W .

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Proof. We know from Theorem 3.5.6 that every root is conjugate to a

simple root. Therefore it will be enough to prove that the simple long

(respectively, short) roots are conjugate. Direct observation of Coxeter

graphs shows that the nodes for any two simple roots of the same length

can be connected by a sequence of edges with marks 3. Hence it will be

enough to prove that if ρi and ρj are distinct simple roots so that mij = 3

then ρi and ρj are conjugate.

Since the system of simple roots is linearly independent, the set ¦ =

¦ © (Zρi + Zρj ) consists of those vectors in ¦ which are linear combinations

of ρi and ρj . We have already checked, on several occasions, that ¦ is

a root system and { ρi , ρj } is a simple system in ¦ . The corresponding

re¬‚ection group W is a dihedral group of order 6. One can see immediately

from a simple diagram that all roots in ¦ form a single W -orbit (check

this!). Alternatively, we can argue as follows1 : re¬‚ections in W are in one-

to-one correspondence with the 3 pairs of opposite roots in ¦ . But W

contains exactly 3 involutions, hence every involution in W is a re¬‚ection.

Every re¬‚ection in W generates a subgroup of order 2 which is a Sylow 2-

subgroup in W . By Sylow™s Theorem, all Sylow 2-subgroups are conjugate

in W , hence all re¬‚ections are conjugate in W , hence all pairs of opposite

roots in ¦ are conjugate, hence all roots in ¦ are conjugate in W and,

since W < W , in W .

Theorem 4.5.2 The orders of the indecomposable re¬‚ection groups are

given in the following table.

|An | = n!

2n · n!

|BCn | =

2n’1 · n!

|Dn | =

27 · 34 · 5

|E6 | =

210 · 34 · 5 · 7

|E7 | =

214 · 35 · 52 · 7

|E8 | =

27 · 32

|F4 | =

|G2 | = 12

Proof. In all cases the highest root ρ0 is a long root. Since all long roots

are conjugate, «

number

|W | = of long · |CW (ρ0 )|.

roots

1

Which is, essentially, part of the solution to Exercise ??

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 93

On the other hand, CW (ρ0 ) = CW (’ρ0 ). One can easily check, using the

formulae of Section 4.4, that (ρ0 , ρi ) 0 for all simple roots ρi , hence

(’ρ0 , ρi ) 0 and the root ’ρ0 belons to the closed fundamental chamber

C. By Theorem 3.7.1, the isotropy group CW (’ρ0 ) is generated by the

simple re¬‚ections which ¬x the root ’ρ0 . These simple re¬‚ections are

exactly the re¬‚ections for the black nodes on the extended Coxeter graphs

in Section 4.4 which are not connected by an edge to the white node ’ρ0 .

Therefore the extended Coxeter graph, with the node ’ρ0 and the nodes

adjacent to ’ρ0 deleted, is the Coxeter graph for W = CW (ρ0 ), which

allows us to determine the isomorphism type and the order of W .

The rest is a case-by-case analysis.

An . We know that |A1 | = 2 = 2! and |A2 | = 6 = 3!. We want to prove

by induction that |An | = (n + 1)!. ¦ contains n(n + 1) roots (all of them

of the same length), and W is of type An’2 . By the inductive assumption,

|W | = [(n ’ 2) + 1]! = (n ’ 1)! and

|W | = n(n + 1) · (n ’ 1)! = (n + 1)!.

Of cource, we know that W = Symn+1 , and there was no much need

in a new proof of the fact that |Symn | = n!. But we wished to use an

opportunity to show how much information about a re¬‚ection group is

contained in its extended Coxeter graph.

BCn . We know that the root systems Bn and Cn have the same mirror

system and re¬‚ection group. It will be more convenient for us compute with

the root system Cn . It contains 2n long roots, and the Coxeter graph for

W is of type Cn’1 . Thus

|W | = 2n · 2n’1 (n ’ 1)! = 2n n!.

Dn . All roots are long, and their number is 2n(n ’ 1). The group W

has disconnected Coxeter graph with connected components of types Dn’2

and A1 , hence W = W — W , where W is of type Dn’2 has, by the

inductive assumption, order 2n’3 (n ’ 2)!, and |W | = 2. Therefore

|W | = 2n(n ’ 1) · (2n’3 (n ’ 2)! · 2) = 2n’1 n!.

E6 . There are 72 roots in ¦, all of them long; the isotropy group W

is of type A5 . Therefore

|W | = 72 · (5 + 1)! = 72 · 6! = 27 · 34 · 5.

E7 . There are 126 roots in ¦, all of them long; the isotropy group W

is of type D6 and has order 25 · 6!. Therefore

|W | = 126 · 25 · 6! = 210 · 34 · 5 · 7.

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E8 . There are 240 roots in ¦, all of them long; the isotropy group W

is of type E7 and has order 210 · 34 · 5 · 7 (just computed). Therefore

|W | = 240 · 210 · 34 · 5 · 7 = 214 · 35 · 52 · 7.

F4 . There are 24 long roots; the isotropy group W is of type C3 and

has order 23 · 3!. Therefore

|W | = 24 · 23 · 3! = 27 · 32 .

Exercises

4.5.1 Prove that the roots in the root systems H3 and H4 form a single orbit

under the action of the corresponding re¬‚ection groups.

4.5.2 Lemma 4.5.1 is not true when the root system ¦ is not indecomposable.

Give an example.

4.5.3 Give an example of a root system of type A1 — A1 — A1 with roots of

three d¬erent lengths.

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[Ber] M. Berger, G´om´trie, Nathan, 1990.

e e

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[G] B. Gr¨nbaum, Convex Polytopes, Interscience Publishers, New

u

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[GB] L. C. Grove, C. T. Benson, Finite Re¬‚ection Groups, Springer-

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[H] J. E. Humphreys, Re¬‚ection Groups and Coxeter Groups,

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[Roc] R. T. Rockafellar, Convex Analysis, Princeton Univerity Press,

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[Ron] M. Ronan, Lectures on Buildings, Academic Press, Boston,

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