Obviously the hyperoctahedron is the dual polytope to the unit cube (picture

(b)).

Vice versa, if is an admissible ordering, then the permutation

n— (n ’ 1)— . . . 1— 1 ... n ’ 1 n

w=

j1 j2 . . . jn jn+1 . . . j2n’1 j2n

where

j1 j2 ... j2n’1 j2n ,

w

is admissible and the ordering coincides with .

Root systems Cn and Bn . Let i , i ∈ [n], be the standard orthonormal

basis in Rn , and set i— = ’ i for i— ∈ [n]— . This de¬nes the vectors j

for all j ∈ [n] [n]— . Now the root system ¦ of type Cn is formed by

the vectors 2 j , j ∈ [n] [n]— (called long roots), together with the vectors

— —

j1 ’ j2 , where j1 , j2 ∈ [n] [n] , j1 = j2 or j2 (called short roots). Written

in the standard basis 1 , 2 , . . . , n , the roots take the form ±2 i or ± i ± j ,

i, j = 1, 2, . . . , n, i = j. Notice that both short and long roots can be

written as j ’ i for some i, j ∈ [n] [n]— .

It is easy to see that when ρ is one of the long roots ±2 i , i ∈ [n], then

sρ is the linear transformation corresponding to the element (i, i— ) of W

in its canonical representation. Analogously, if ρ = i ’ j , is a short root

(recall that we use the convention i— = ’ i for i ∈ [n]), then the re¬‚ection

sr corresponds to the admissible permutation (i, j)(i— , j — ). Moreover, one

can easily check (for example, by computing the eigenvalues of admissible

permutations from W in their action on Rn ) that every re¬‚ection in the

58

d

T d

C2 B2

d d

d d

d d

s

d s T d

d d

d d

'

d E

d ' d E d

d d d d

d d d d

d c d

d

© ‚ d

© ‚

d d

d d

d

c d

Figure 2.17: Root systems B2 and C2 .

group of the symmetries of the unit cube [’1, 1]n is of one of these two

types.

Now we see that use of the name ˜root system™ in regard to the set ¦

is justi¬ed.

The root system

Bn = { ± i ± j , ± i | i, j = 1, 2 . . . , n, i = j }

di¬ers from Cn only in lengths of roots (see Figure 2.17) and has the same

re¬‚ection group BCn . Therefore in the sequel we shall deal only with the

root system Cn .

Action of W on ¦. Now consider the linear functional

f (x) = x1 + 2x2 + 3x3 + · · · + nxn .

It is easy to see that a root i ’ j is positive with respect to f if, in the

ordering

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

of the set [n] [n]— , we have i > j. The system of positive roots ¦+

associated with f is called the standard positive system of roots. The set

Π = {2 1 , ’ 1, . . . , n ’ n’1 }

2

is obviously the simple system of roots contained in ¦+ .

If now

j1 <w j2 <w . . . <w j2n’1 <w j2n ,

is an admissible ordering of [n] [n]— , then the vectors jn+1 , jn+2 , . . . , j2n

form a basis in Rn . Let y1 , y2 , . . . , yn be the coordinates with respect to

this basis and f (y) = y1 +2y2 +3y3 +· · ·+nyn . Then, obviously, f does not

vanish on roots in ¦, and, for a root j ’ i in ¦, the inequality f ( j ’ i ) > 0

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 59

is equivalent to i w j. Thus the system of positive roots associated with

f coincides with the system

w¦+ = { j ’ w

|i j}

i

obtained from the standard system ¦+ of positive roots by the action of

the element w. Obviously, the simple system of roots contained in ¦+ is

exactly wΠ.

If now Π is an arbitrary simple system of roots arising from an arbitrary

linear function f : Rn ’’ R not vanishing on roots in ¦ then the following

objects are uniquely determined by our choice of Π :

• the system of positive roots ¦+ , which can be de¬ned in two equiv-

alent ways: as the set of all roots which are non-negative linear com-

binations of roots from Π , and as the set {r ∈ ¦ | f (r) > 0};

• the (obviously admissible) ordering on J de¬ned by the rule: i j

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

In particular, we immediately have the following observation (which is a

partial case of a more general result about conjugacy of simple system of

roots for arbitrary ¬nite re¬‚ection groups, Theorem 3.5.5).

Proposition 2.8.1 Any two simple systems in the root system ¦ of type

Bn or Cn are conjugate under the action of W . Moreover, the re¬‚ection

group W is simply transitive in its action on the set of simple systems in

¦.

Exercises.

2.8.1 Prove that every re¬‚ection in BCn has the form (ii— ) or (ij)(i— j — ).

2.8.2 Prove that the re¬‚ection group of type BC2 is isomorphic to the dihedral

group D8 .

2.8.3 The group of symmetries of the cube. Observe that the group

W = BC3 of symmetries of the cube ∆ = [’1, 1]3 contains the involution z

which sends every vertex of the cube to its opposite.

1. Check that det z = ’1, so that z to the group R = Rot ∆ of rotations of

the cube.

2. Prove that z ∈ Z(W ).

3. Prove that the group R acts faithfully on the set D of 4 diagonals of

the cube ∆, that is, the segments connecting the opposite vertices of the

cube. Moreover, every permutation of diagonals is the result of action of

a rotation of the cube. Hence R Sym4 .

60

4. Prove that W = z — R.

5. Prove that z = Z(W ).

6. Prove that the symmetries of the cube which send every 2-dimensional

face of the cube into itself or the opposite face form a normal abelian

subgroup E < W of order 8. Prove further that W/E Sym3 and that

actually W = E T for some subgroup T Sym3 .

2.8.4 Important root subsystems.

Prove that

1. the set ˜ of roots { ± i | i = 1, . . . , n } is a root system of type A1 +· · ·+A1

(n summands);

2. the intersection Ψ of ¦ with the hyperplane x1 + · · · + xn = 0 is a root

system of type An’1 .

2.8.5 The structure of the hyperoctahedral group. Use Exercise 2.8.4

to show that if E and T are the re¬‚ection groups corresponding to the systems

of roots ˜ and Ψ then

Z2 — · · · — Z2 (n factors);

1. E

2. E W;

3. T Symn ;

4. W = E T.

2.8.6 (R. Sandling) Prove that

Sym [’1, 1]n = {w ∈ GLn (Rn ) | w([’1, 1]n ) = [’1, 1]n },

i.e. linear transformations preserving the cube are in fact orthogonal.

2.9 The root system Dn

By de¬nition,

Dn = { ± i ± | i, j = 1, 2, . . . , n, i = j };

j

thus Dn is a subsystem of the root system Cn .

The system

Π={ + 2, 2 ’ 1, 3 ’ 2, . . . , n ’ }

1 n’1

is a simple system in ¦.

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 61

—

¡ ——

¡ # ¡ ¡ ¡ (¡

——

¡ ¡ ¡ ¡ ¡ (¡

—— ¡

¡d

s

d (¡

¡ ¡ — ¡

dg

y d g d

g

(

g

ydg X d gd (

d g d g gd (

d g d g g d(

W z g ¨¨

gd d g d

d

gd

g g¨

—— d

—

d¡

¡ ¡ ¡ ¡

—— d

d

© ‚

¡ ¡ d¡ ¡ ¡

——d

d¡ d¡

¡ ¡ ¡ —

Figure 2.18: Shown are the root system D3 inscribed into the unit cube [’1, 1]3

(on the left), and the corresponding mirror system (shown in the middle by

intersections with the surface of the cube and the tetrahedron inscribed in the

cube). Comparing the last two pictures we see that the mirror system of type

D3 is isometric to the mirror system of type A3 .

Exercises

2.9.1 Describe explicitly an isometry between the root systems

D3 = { ± i ± | i, j = 1, 2, 3, i = j }

j

and

A3 = { ’ | i, j = 1, 2, 3, 4, i = j }

i j

(see Figure 2.18).

2.9.2 Sketch the root system D2 ; you will see that it consists of two orthogonal

pairs of vectors, each forming the 1-dimensional system A1 . Thus D2 = A1 +A1 .

62

Chapter 3

Coxeter Complex

3.1 Chambers

Chambers. Consider the system Σ of all mirrors of re¬‚ections sρ for