Coxeter
group
|
Coxeter
diagram
|
Degrees
|
Coxeter planes
|
| A2
|
 
|
2, 3
|
A1, A2
|
| B2
|
|
2, 4
|
A1, B2
|
| H2
|
|
2, 5
|
A1, H2
|
| A3
|
|
2, 3, 4
|
A1, A2, A3
|
| B3
|
|
2, 4, 6
|
A1, B2, A2=B3
|
| H3
|
|
2, 6, 10
|
A1, A2, H2=H3
|
| A4
|
|
2, 3, 4, 5
|
A1, A2, A3, A4
|
| B4
|
|
2, 4, 6, 8
|
A1, A3, B2, A2=B3, B4
|
| D4
|
 
|
2, 4, 6
|
A1, A3, A2=D4
|
| F4
|
|
2, 6, 8, 12
|
A1, A3=B2, A2=B3, F4
|
| H4
|
|
2, 12, 20, 30
|
A1, A2, A3, H2=H3, H4
|
| A5
|
|
2, 3, 4, 5, 6
|
A1, A2, A3, A4, A5
|
| B5
|
|
2, 4, 6, 8, 10
|
A1, A3=B2, A2=B3, B4, A4=B5
|
| D5
|
|
2, 4, 6, 8; 5
|
A1, A3, A2=D4, D5; A4
|
| A6
|
|
2, 3, 4, 5, 6, 7
|
A1, A2, A3, A4, A5, A6
|
| B6
|
|
2, 4, 6, 8, 10, 12
|
A1, A3=B2, A2=B3, B4, A4=B5, B6
|
| D6
|
|
2, 4, 6, 8, 10
|
| E6
|
 
|
2, 5, 6, 8, 9, 12
|
A1, A4, A2=D4=A5, A3=D5, ?, E6
|
| E7
|
  
|
2, 6, 8, 10, 12, 14, 18
|
| E8
|
|
2, 8, 12, 14, 18, 20, 24, 30
|
Finite Coxeter group foldings
Let me try using Coxeter–Dynkin_diagram#Geometric_foldings to express Coxeter planes as Coxeter numbers and all degrees of fundamental invariants. Foldings are shown by marking node with colors, re and blue, which map to node 1 or 2 in the rank 2 folded group.
Example: A3,
| Folding |
Degree |
Coxeter Plane
|
  |
|
4 |
A3
|
|
|
3 |
A2
|
|
|
2 |
A1
|
Example: B3,
| Folding |
Degree |
Coxeter Plane
|
|
 |
6 |
B3
|
|
|
3×2 |
A2
|
|
|
4 |
B2
|
|
|
2 |
A1
|
Example: H3,
| Folding |
Degree |
Coxeter Plane
|
|
 |
10 |
H3
|
|
|
5×2 |
H2
|
|
|
3×2 |
A2
|
|
|
2 |
A1
|
Example: A4,
| Folding |
Degree |
Coxeter Plane
|
|
|
|
5 |
A4
|
|
|
|
4 |
A3
|
|
|
|
3 |
A2
|
|
|
|
2 |
A1
|
Example: B4,
| Folding |
Degree |
Coxeter Plane
|
|
|
 |
8 |
B4
|
|
|
|
6 |
B3
|
|
|
|
3×2 |
A2
|
|
|
|
4 |
A3
|
|
|
|
4 |
B2
|
|
|
|
2 |
A1
|
Example: D4,
| Folding |
Degree |
Coxeter Plane
|
 |
|
|
6 |
D4=B3
|
 |
|
|
3×2 |
A2
|
|
= |
|
4 |
D3=A3
|
|
|
|
4 |
B2
|
|
|
|
2 |
A1
|
Example: F4,
| Folding |
Degree |
Coxeter Plane
|
|
|
 |
12 |
F4
|
|
|
|
4×2 |
A3
|
|
|
|
4×2 |
B2
|
|
|
|
6 |
B3
|
|
|
|
3×2 |
A2
|
|
|
|
2 |
A1
|
Example: H4,
| Folding |
Degree |
Coxeter Plane
|
|
|
  |
30 |
H4
|
|
|
 |
20 |
|
|
|
|
12 |
F4
|
|
|
|
10 |
H3
|
|
|
|
5×2 |
H2
|
|
|
|
3×2 |
A2
|
|
|
|
4 |
A3
|
|
|
|
2 |
A1
|
Example: A5,
| Folding |
Degree |
Coxeter Plane
|
|
|
|
6 |
A5
|
|
|
|
5 |
A4
|
|
|
|
4 |
A3
|
|
|
|
3 |
A2
|
|
|
|
2 |
A1
|
Example: B5,
| Folding |
Degree |
Coxeter Plane
|
|
|
|
10 |
B5
|
|
|
|
5×2 |
A4
|
|
|
|
8 |
B4
|
|
|
|
6 |
B3
|
|
|
|
3×2 |
A2
|
|
|
|
4 |
A3
|
|
|
|
4 |
B2
|
|
|
|
2 |
A1
|
Example: D5,
| Folding |
Degree |
Coxeter Plane
|
|
|
|
8 |
D5=B4
|
|
= |
|
6 |
D4=B3
|
 |
|
|
3×2 |
A2
|
|
|
|
5 |
A4
|
|
= |
|
4 |
D3=A3
|
|
|
|
2 |
A1
|
Example: E6,
| Folding |
Degree |
Coxeter Plane
|
|
|
|
12 |
E6 = F4
|
|
|
 |
9 |
|
 |
= |
|
8 |
D5 = B4
|
|
|
|
6 |
A5
|
|
= |
|
6 |
D4 = B3
|
|
|
|
3×2 |
A2
|
|
|
|
5 |
A4
|
|
|
|
4 |
A3
|
|
|
|
2 |
A1
|
