¢ d E

h vertices

¢ d h x1

d

¢ x1 = x2

(1, 0, 0), (0, 1, 0), (0, 0, 1).

¢

(0, 1, 0)

x

2

©

Figure 2.14: Symn is the group of symmetries of the regular simplex.

’

(’1, ’1, 1) (’1, 1, 1)

3 1

¡ # ¡

’ 2¡ ¡

3

¡s

d

d The root system { i ’ j | i = j }

¡

(1, ’1, 1) (1, 1, 1)

d of type A2 lies in the hyperplane

X 2’ 1

d

d x1 + x2 + x3 = 0 which cuts a regular

d

hexagon in the unit cube [’1, 1]3 .

W

’

1 2 d

(’1, ’1, ’1)

d

d ¡ (’1, 1, ’1)

¡ ‚ 2’ 3

¡ ¡

¡ ¡(1, 1, ’1)

’

(1, ’1, ’1) 1 3

Figure 2.15: Root system of type A2 .

W = W (¦) is the corresponding re¬‚ection group, and the mirror system

Σ consists of all hyperplanes xi = xj , i = j, i, j = 1, . . . , n.

Notice that the group W is not essential for V ; indeed, it ¬xes all

points in the 1-dimensional subspace R( 1 +· · ·+ n ) and leaves invariant the

(n’1)-dimensional linear subpace U de¬ned by the equation x1 +· · ·+xn =

0. It is easy to see that ¦ ‚ U spans U . In particular, the rank of the root

system ¦ is n, which justi¬es the use, in accordance with our convention,

of the index n ’ 1 in the notation An’1 for it.

The standard simple system. Take the linear functional

f (x) = x1 + 2x2 + · · · + nxn .

54

Obviously f does not vanish on roots, and the corresponding positive sys-

tem has the form

¦+ = { i ’ j | j < i }.

The set of positive roots

Π={ ’ 1, 3 ’ 2, . . . , n ’ }

2 n’1

is linearly independent and every positive root is obviously a linear combi-

nation of roots in Π with nonnegative coe¬cients: for i > j,

’ =( i’ i’1 ) + ··· + ( ’ j ),

i j j+1

Therefore Π is a simple system. It is called the standard simple system of

the root system An’1 .

Action of Symn on the set of all simple systems. The following

result is a partial case of Theorem 3.5.1. But the elementary proof given

here is instructive on its own.

Lemma 2.7.2 The group W = Symn acts simply transitively on the set

of all positive (resp. simple) systems in ¦.

Proof. Since there is a natural one-to-one correspondence between simple

and positive systems, it is enough to prove that W acts simply transitivly

on the set of positive systems in ¦.

Let f be an arbitrary linear functional which does not vanish on ¦,

that is, f ( i ’ j ) = 0 for all i = j. Then all the values

f ( 1 ), . . . , f ( n )

are di¬erent and we can list them in the strictly increasing order:

f( i1 ) < f( i2 ) < . . . < f( in ).

Now consider the permutation w given, in the column notation, as

1 2 ··· n ’ 1 n

w= .

i1 i2 · · · in’1 in

w

Thus the functional f de¬nes a new ordering, wich we shall denote as ,

on the set [n]:

j w i if and only if f ( j ) f ( i ).

If we look again at the table for w we see that above any element i in the

bottom row lies, in the upper row, the element w’1 i. Thus

w

j if and only if w’1 i w’1 j.

i

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 55

Notice also that the permutation w and the associated ordering w of [n]

uniquely determine each other6 .

Now consider the positive system ¦+ de¬ned by the functional f ,

0

¦+ = { ’ | f ( i ’ j ) > 0 }.

i j

0

We have the following chain of equivalences:

∈ ¦+

’ i¬ f ( j ) < f ( i)

i j 0

j <w i

i¬

w’1 j < w’1 i

i¬

+

i¬ w’1 i ’ w’1 j ∈ ¦

w’1 ( i ’ j ) ∈ ¦+

i¬

+

i¬ i ’ j ∈ w¦ .

This proves that ¦+ = w¦+ and also that the permutation w is uniquely

0

determined by the positive system ¦+ . Since ¦+ , by its construction from

0

an arbitrary functional, represents an arbitrary positive system in ¦, the

group W acts on the set of positive systems in ¦ simply transitively.

Exercises

2.7.1 Make a sketch of the root systems A1 • A1 in R2 and A1 • A1 • A1 in

R3 .

2.7.2 Check that, when we take the intersections of the mirrors of re¬‚ections

in W = Symn to the subspace x1 + · · · + xn = 0 of Rn , the resulting system of

mirrors can be geometrically described as the system of mirrors of symmetry of

the regular (n ’ 1)-simplex with the vertices

1

δi = ’ ( + ··· + n ), i = 1, . . . , n.

i 1

n

2.7.3 An orthogonal transformation of the Euclidean space Rn is called a rota-

tion if its determinant is 1. For a polytope ∆, denote by Rot ∆ the subgroup of

Sym ∆ formed by all rotations. We know that the group of symmetries Sym ∆

of the regular tetrahedron ∆ in R3 is isomorphic to Sym4 ; prove that Rot ∆ is

the alternating group Alt4 .

6

In the nineteenth century the orderings, or rearrangements, of [n] were called per-

mutations, and the permutations in the modern sense, i.e. maps from [n] to [n], were

called substitutions. These are two aspects, ˜passive™ and ˜active™, of the same object.

We shall see later that they correspond to treating a permutation as an element of a

re¬‚ection group Symn or an element of the Coxeter complex for Symn .

56

2.8 Root systems of type Cn and Bn

Hyperoctahedral group. Let

[n] = {1, 2, . . . , n} and [n]— = {1— , 2— , . . . , n— }.

De¬ne the map — : [n] ’’ [n]— by i ’ i— and the map — : [n]— ’’ [n] by

(i— )— = i. Then — is an involutive permutation7 of the set [n] [n]— .

Let W be the group of all permutations of the set [n] [n]— which com-

mute with the involution — , i.e. a permutation w belongs to W if and only

if w(i— ) = w(i)— for all i ∈ [n] [n]— . We shall call permutations with this

property admissible. The group W is known under the name of hyperoc-

tahedral group BCn . It is easy to see that W is isomorphic to the group

of symmetries of the n-cube [’1, 1]n in the n-dimensional real Euclidean

space Rn . Indeed, if 1 , 2 , . . . , n is the standard orthonormal basis in Rn ,

we set, for i ∈ [n], i— = ’ i . Then we can de¬ne the action of W on Rn

by the following rule: w i = wi . Since w is an admissible permutation of

[n] [n]— , the linear transformation is well-de¬ned and orthogonal. Also it

can be easily seen that W is exactly the group of all orthogonal transfor-

mations of Rn preserving the set of vectors {± 1 , ± 2 , . . . , ± n } and thus

preserving the cube [’1, 1]n . Indeed, the vectors ± i , i ∈ [n], are exactly

the unit vectors normal to the (n ’ 1)-dimensional faces of the cube (given,

obviously, by the linear equations xi = ±1, i = 1, 2, . . . , n).

The name ˜hyperoctahedral™ for the group W is justi¬ed by the fact

that the group of symmetries of the n-cube coincides with the group of

symmetries of its dual polytope, whose vertices are the centers of the faces

of the cube. The dual polytope for the n-cube is known under the name

of n-cross polytope or n-dimensional hyperoctahedron (see Figure 2.16).

Admissible orderings. We shall order the set [n] [n]— in the following

way:

n— < n ’ 1— < ... < 2— < 1— < 1 < 2 < ... < n ’ 1 < n.

w

[n]— by

If now w ∈ W then we de¬ne a new ordering of the set [n]

the rule

w

j if and only if w’1 i w’1 j.

i

Orderings of the form w , w ∈ W , are called admissible orderings of the

set [n] [n]— . They can be characterized by the following property:

[n]— is admissible if and only if from

an ordering on [n]

i j it follows that j — i— .

7

That is, a permutation of order 2.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 57

¡ ¡

3

¡ ¡

¡ ¡

T d

g d

d

gd ¡ ¡

gd d

’

g— 1 d

d

¡ ——d

! d

—E d

'

¡ d

—

’ 2— ——

2

d—— ¡ d—

—— ¡ d ——

d

d 1g g

d

dg dg

dg

¡ ¡

dg

¡ ¡

g g

dc d

’ ¡ ¡

3

¡ ¡

(a) (b)

Figure 2.16: Hyperoctahedron (˜octahedron™ in dimension n = 3) or n-cross