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n 3 look di¬erently.


4 4
s s c
ρ1 ρ2 ’ρ0
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 87

Extended Coxeter diagram for B2




’ρ0
c
4
s. . . s
s s
r
ρ3 ρn’1r s
ρ1 ρ2 r
ρn




Extended Coxeter diagram for Bn , n 3.


be the standard basis in Rn .
Root system Cn , n 2. Let 1, . . . , n


¦ = { ±2 i , ± i ± ej | i, j = 1, . . . , n, i < j },
Π = { 2 1 , 2 ’ 1 , . . . , n ’ n’1 }.

¦ contains 2n long roots ±2 i and 2n(n ’ 1) short roots ± i ± j , i < j.
We enumerate the simple roots as

ρ1 = 2 1 , ρ2 = ’ 1, . . . , ρn = ’ n’1 .
2 n


The highest root is
ρ0 = 2 n .



4 4
c s s
’ρ0 ρ1 ρ2




Extended Coxeter diagram for C2




4 4
s. . . s
s s s c
ρ1 ρ2 ρ3 ρn’1 ρn ’ρ0




Extended Coxeter diagram for Cn , n 3.
88

be the standard basis in Rn .
Root system Dn , n 4. Let 1, . . . , n


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


thus Dn is a subsystem of the root system Cn . All roots have the same
length. The total number of roots is 2n(n ’ 1).
The simple system Π is

ρ1 = + 2, ρ2 = ’ 1, ρ3 = ’ 2, . . . , ρn = ’
1 2 3 n n’1


The highest root is
ρ0 = + n.
n’1




ρ1 ’ρ0
s c
r
rs s. . . s
r s
¨¨
r
ρn’2 ρn’1r s
¨ ρ3 r
s
ρ2 ρn




be the standard basis in R8 .
Root system E8 . Let 1, . . . , 8

8
1
¦= ± i ± ej (i < j), ± (even number of + signs) ,
i
2 i=1

for Π take
1
ρ1 = ( 1’ 2’ 3’ 4’ 5’ ’ + 8 ),
6 7
2
ρ2 = 1 + 2,
ρi = i’1 ’ i’2 (3 i 8).

All roots have the same length; the total number of roots is 240.
The highest root is
ρ0 = 7 + 8 .


ρ1 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ’ρ0
s s s s s s s c

s
ρ2
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 89

Root system E7 . Take the root system of type E8 in R8 just constructed
and consider the span V of the roots ρ1 , . . . , ρ7 . Let ¦ be the set of 126
roots of E8 belonging to V :

± i± (1 i<j 6),
j


±( ’ 8 ),
7
6
1
± ’ + ± ,
7 8 i
2 i=1

where the number of minus signs in the sum is odd.
All roots have the same length. The roots

ρ1 , . . . , ρ7

form a simple system, and the highest root is

ρ0 = ’ 7.
8




’ρ0 ρ1 ρ3 ρ4 ρ5 ρ6 ρ7
c s s s s s s

s
ρ2




Root system E6 . Again we start with the root system of type E8 in R8 .
Denote by V the span of the roots ρ1 , . . . , ρ6 , and take for ¦ the 72 roots
of E8 belonging to V :

± i± (1 i<j 5),
j

5
1
± ’ ’ + ± ,
8 7 6 i
2 i=1

where the number of minus signs in the sum is odd.
All roots have the same length. The roots

ρ1 , . . . , ρ6

form a simple system, and the highest root is
1
ρ0 = ( + + + + ’ ’ + 8 ).
1 2 3 4 5 6 7
2
90



ρ1 ρ3 ρ4 ρ5 ρ6
s s s s s

sρ2

c
’ρ0




be the standard basis in R4 . ¦ consists
Root system F4 . Let 1, . . . , 4
of 24 long roots
± i± (i < j)
j

and 24 short roots
1
± i, (± ± ± ± 4 ).
1 2 3
2
For a simple system Π take
1
ρ1 = ’ 3, ρ2 = ’ 4, ρ3 = 4, ρ4 = ( ’ ’ ’ 4 ).
2 3 1 2 3
2
The highest root is
ρ0 = + 2.
1




4
c s s s s
ρ0 ρ1 ρ2 ρ3 ρ4




Root system G2 . Let V be the hyperplane x1 + x2 + x3 = 0 in R3 . ¦
consists of 6 short roots

±( i ’ j ), i < j,

and 6 long roots
±(2 i ’ ’ k ),
j

where i, j, k are all di¬erent. For a simple system Π take

ρ1 = ’ 2, ρ2 = ’2 + + 3.
1 1 2


The highest root is
ρ0 = + 2.
1
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 91



6
s s c
ρ1 ρ2 ’ρ0

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