User:Tomruen/Coxeter foldings: Difference between revisions
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|{{CDD|node_c2|3|node_c3|split1|nodeab_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2 ||A<sub>2</sub> |
|{{CDD|node_c2|3|node_c3|split1|nodeab_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2 ||A<sub>2</sub> |
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|- align=center BGCOLOR="#ffe0e0" |
|- align=center BGCOLOR="#ffe0e0" |
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|{{CDD|node_c2|3|node_c2|split1|nodeab_c3}} ||{{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4|| |
|{{CDD|node_c2|3|node_c2|split1|nodeab_c3}} || {{CDD|node_c2|split1|nodeab_c3}} = {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||D<sub>3</sub>=A<sub>3</sub> |
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|- align=center BGCOLOR="#ffe0e0" |
|- align=center BGCOLOR="#ffe0e0" |
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| || || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub> |
| || || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub> |
||
Revision as of 23:36, 29 October 2017
| 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 | |
| E7 |   
|
2, 6, 8, 10, 12, 14, 18 | |
| E8 |
|
2, 8, 12, 14, 18, 20, 24, 30 |
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
| Folding | Degree | Coxeter Plane | |
|---|---|---|---|
  |
|
4 | A3 |
|
|
3 | A2 |
|
|
2 | A1 |
B3
| Folding | Degree | Coxeter Plane | |
|---|---|---|---|
|
 |
6 | B3 |
|
|
3×2 | A2 |
|
|
4 | B2 |
|
|
2 | A1 |
H3
| Folding | Degree | Coxeter Plane | |
|---|---|---|---|
|
 |
10 | H3 |
|
|
5×2 | H2 |
|
|
3×2 | A2 |
|
|
2 | A1 |
A4
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
|
5 | A4 | |
|
|
|
4 | A3 |
|
|
3 | A2 | |
|
|
2 | A1 | |
B4
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
 |
8 | B4 | |
|
|
|
6 | B3 |
|
|
3×2 | A2 | |
|
|
|
4 | A3 |
|
|
4 | B2 | |
|
|
2 | A1 | |
D4
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
 |
|
|
6 | D4=B3 |
 |
|
3×2 | A2 | |
|
= |
|
4 | D3=A3 |
|
4 | B2 | ||
|
|
2 | A1 | |
F4
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
 |
12 | F4 | |
|
|
|
6 | B3 |
|
|
3×2 | A2 | |
|
|
|
4 | A3 |
|
|
4 | B2 | |
|
|
2 | A1 | |
H4
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
  |
30 | H4 | |
 |
20 | |||
|
12 | F4 | ||
|
|
|
10 | H3 |
|
|
5×2 | H2 | |
|
|
3×2 | A2 | |
|
|
|
4 | A3 |
|
|
2 | A1 | |
A5
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
|
6 | A5 | |
|
|
|
5 | A4 |
|
|
|
4 | A3 |
|
|
3 | A2 | |
|
|
2 | A1 | |
B5
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
|
10 | B5 | |
|
|
|
5×2 | A4 |
|
|
|
8 | B4 |
|
|
|
6 | B3 |
|
|
3×2 | A2 | |
|
|
|
4 | A3 |
|
|
4 | B2 | |
|
|
2 | A1 | |
D5
| Folding | Degree | Coxeter Plane | ||
|---|---|---|---|---|
|
|
|
8 | D5=B4 |
|
= |
|
6 | D4=B3 |
 |
|
3×2 | A2 | |
|
|
|
5 | A4 |
|
|
= |
|
4 | D3=A3 |
|
|
2 | A1 | |