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h d (1, 0, 0)
¢ 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

w’1 j < w’1 i

+
i¬ w’1 i ’ w’1 j ∈ ¦
w’1 ( i ’ j ) ∈ ¦+

+
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

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