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User:Tomruen/Coxeter foldings

From Wikipedia, the free encyclopedia
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.

A3

[edit]
Example: A3,
Folding Degree Coxeter Plane
4 A3
3 A2
2 A1

B3

[edit]
Example: B3,
Folding Degree Coxeter Plane
6 B3
3×2 A2
4 B2
2 A1

H3

[edit]
Example: H3,
Folding Degree Coxeter Plane
10 H3
5×2 H2
3×2 A2
2 A1

A4

[edit]
Example: A4,
Folding Degree Coxeter Plane
5 A4

4 A3

3 A2
2 A1

B4

[edit]
Example: B4,
Folding Degree Coxeter Plane
8 B4

6 B3

3×2 A2
4 A3
4 B2
2 A1

D4

[edit]
Example: D4,
Folding Degree Coxeter Plane
6 D4=B3
3×2 A2
= 4 D3=A3
4 B2
2 A1

F4

[edit]
Example: F4,
Folding Degree Coxeter Plane
12 F4
4×2 A3
4×2 B2
6 B3
3×2 A2
2 A1

H4

[edit]
Example: H4,
Folding Degree Coxeter Plane
30 H4
20
12 F4

10 H3
5×2 H2

3×2 A2
4 A3
2 A1

A5

[edit]
Example: A5,
Folding Degree Coxeter Plane
6 A5

5 A4



4 A3
3 A2
2 A1

B5

[edit]
Example: B5,
Folding Degree Coxeter Plane
10 B5
5×2 A4
8 B4


6 B3


3×2 A2


4 A3
4 B2
2 A1

D5

[edit]
Example: D5,
Folding Degree Coxeter Plane
8 D5=B4

= 6 D4=B3


3×2 A2
5 A4



= 4 D3=A3
2 A1

E6

[edit]
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