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{| class="wikitable" align=right
{| class="wikitable" align=right width=420
![[Coxeter group|Coxeter<BR>group]]
![[Coxeter group|Coxeter<BR>group]]
![[Coxeter diagram|Coxeter<BR>diagram]]
!width=75|[[Coxeter diagram|Coxeter<BR>diagram]]
! Degrees
! Degrees
!Coxeter planes
|- align=center
! A<sub>2</sub>
|{{CDD|node|3|node}}
| 2, 3
|A<sub>1</sub>, A<sub>2</sub>
|- align=center
! B<sub>2</sub>
| {{CDD|node|4|node}}
| 2, 4
|A<sub>1</sub>, B<sub>2</sub>
|- align=center
! H<sub>2</sub>
|{{CDD|node|5|node}}
| 2, 5
|A<sub>1</sub>, H<sub>2</sub>
|- align=center
|- align=center
! A<sub>3</sub>
! A<sub>3</sub>
|{{CDD|node|3|node|3|node}}
|{{CDD|node|3|node|3|node}}
| 2, 3, 4
| 2, 3, 4
|A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>
|- align=center
! A<sub>4</sub>
|{{CDD|node|3|node|3|node|3|node}}
| 2, 3, 4, 5
|- align=center
! A<sub>5</sub>
|{{CDD|node|3|node|3|node|3|node|3|node}}
| 2, 3, 4, 5, 6
|- align=center
! A<sub>''n''</sub>
|{{CDD|node|3|node|3}}...{{CDD|3|node|3|node}}
| 2, 3, 4, ..., ''n'' + 1
|- align=center
|- align=center
! B<sub>3</sub>
! B<sub>3</sub>
| {{CDD|node|4|node|3|node}}
| {{CDD|node|4|node|3|node}}
| 2, 4, 6
| 2, 4, 6
|A<sub>1</sub>, B<sub>2</sub>, A<sub>2</sub>=B<sub>3</sub>
|- align=center
! H<sub>3</sub>
|{{CDD|node|5|node|3|node}}
| 2, 6, 10
|A<sub>1</sub>, A<sub>2</sub>, H<sub>2</sub>=H<sub>3</sub>
|- align=center
! A<sub>4</sub>
|{{CDD|node|3|node|3|node|3|node}}
| 2, 3, 4, 5
|A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, A<sub>4</sub>
|- align=center
|- align=center
! B<sub>4</sub>
! B<sub>4</sub>
| {{CDD|node|4|node|3|node|3|node}}
| {{CDD|node|4|node|3|node|3|node}}
| 2, 4, 6, 8
| 2, 4, 6, 8
|A<sub>1</sub>, A<sub>3</sub>, B<sub>2</sub>, A<sub>2</sub>=B<sub>3</sub>, B<sub>4</sub>
|- align=center
! B<sub>''n''</sub>
| {{CDD|node|4|node|3}}...{{CDD|3|node|3|node}}
| 2, 4, 6, ..., 2''n''
|- align=center
|- align=center
! D<sub>4</sub>
! D<sub>4</sub>
|{{CDD|nodes|split2|node|3|node}}
|{{CDD|nodes|split2|node|3|node}}
| 2, 4, 6
| 2, 4, 6
| A<sub>1</sub>, A<sub>3</sub>, A<sub>2</sub>=D<sub>4</sub>
|- align=center
! F<sub>4</sub>
|{{CDD|node|3|node|4|node|3|node}}
| 2, 6, 8, 12
|A<sub>1</sub>, A<sub>3</sub>=B<sub>2</sub>, A<sub>2</sub>=B<sub>3</sub>, F<sub>4</sub>
|- align=center
! H<sub>4</sub>
|{{CDD|node|5|node|3|node|3|node}}
| 2, 12, 20, 30
|A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, H<sub>2</sub>=H<sub>3</sub>, H<sub>4</sub>
|- align=center
! A<sub>5</sub>
|{{CDD|node|3|node|3|node|3|node|3|node}}
| 2, 3, 4, 5, 6
|A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, A<sub>4</sub>, A<sub>5</sub>
|- align=center
! B<sub>5</sub>
| {{CDD|node|4|node|3|node|3|node|3|node}}
| 2, 4, 6, 8, 10
|A<sub>1</sub>, A<sub>3</sub>=B<sub>2</sub>, A<sub>2</sub>=B<sub>3</sub>, B<sub>4</sub>, A<sub>4</sub>=B<sub>5</sub>
|- align=center
|- align=center
! D<sub>5</sub>
! D<sub>5</sub>
|{{CDD|nodes|split2|node|3|node|3|node}}
|{{CDD|nodes|split2|node|3|node|3|node}}
| 5; 2, 4, 6, 8
| 2, 4, 6, 8; 5
|A<sub>1</sub>, A<sub>3</sub>, A<sub>2</sub>=D<sub>4</sub>, D<sub>5</sub>; A<sub>4</sub>
|- align=center
! A<sub>6</sub>
|{{CDD|node|3|node|3|node|3|node|3|node|3|node}}
| 2, 3, 4, 5, 6, 7
|A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, A<sub>4</sub>, A<sub>5</sub>, A<sub>6</sub>
|- align=center
! B<sub>6</sub>
| {{CDD|node|4|node|3|node|3|node|3|node|3|node}}
| 2, 4, 6, 8, 10, 12
|A<sub>1</sub>, A<sub>3</sub>=B<sub>2</sub>, A<sub>2</sub>=B<sub>3</sub>, B<sub>4</sub>, A<sub>4</sub>=B<sub>5</sub>, B<sub>6</sub>
|- align=center
|- align=center
! D<sub>''n''</sub>
! D<sub>6</sub>
|{{CDD|nodes|split2|node|3}}...{{CDD|3|node|3|node}}
|{{CDD|nodes|split2|node|3|node|3|node|3|node}}
| ''n''; 2, 4, 6, ..., 2''n'' &minus; 2
| 2, 4, 6, 8, 10
|- align=center
|- align=center
! E<sub>6</sub>
! E<sub>6</sub>
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|{{CDD|node|3|node|split1|nodes|3ab|nodes}}
| 2, 5, 6, 8, 9, 12
| 2, 5, 6, 8, 9, 12
|A<sub>1</sub>, A<sub>4</sub>, A<sub>2</sub>=D<sub>4</sub>=A<sub>5</sub>, A<sub>3</sub>=D<sub>5</sub>, ?, E<sub>6</sub>
|- align=center
|- align=center
! E<sub>7</sub>
! E<sub>7</sub>
Line 55: Line 101:
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
| 2, 8, 12, 14, 18, 20, 24, 30
| 2, 8, 12, 14, 18, 20, 24, 30
|- align=center
! F<sub>4</sub>
|{{CDD|node|3|node|4|node|3|node}}
| 2, 6, 8, 12
|- align=center
! G<sub>2</sub>
|{{CDD|node|6|node}}
| 2, 6
|- align=center
! H<sub>3</sub>
|{{CDD|node|5|node|3|node}}
| 2, 6, 10
|- align=center
! H<sub>4</sub>
|{{CDD|node|5|node|3|node|3|node}}
| 2, 12, 20, 30
|}
|}
[[File:Geometric_folding_Coxeter_graphs.png|thumb|Finite Coxeter group foldings]]
[[File:Geometric_folding_Coxeter_graphs.png|thumb|Finite Coxeter group foldings]]


Notes - curious if [[Coxeter–Dynkin_diagram#Geometric_folding]]s can be expressed as [[Coxeter_element#Definitions|Coxeter numbers]] and ''all degrees of fundamental invariants'' AND its clear some degrees exist in potentially multiple unique foldings, and in general every branch all by itself can map to the plane.
Let me try using [[Coxeter–Dynkin_diagram#Geometric_folding]]s to express [[Coxeter plane]]s as [[Coxeter_element#Definitions|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.

Foldings can be marked by node colors, red and blue, which map to node 1 or 2 in the rank 2 folded group.
==A3==
==A3==
{| class=wikitable
{| class=wikitable
|+ Example: A<sub>3</sub>, {{CDD|node|3|node|3|node}}
|+ Example: A<sub>3</sub>, {{CDD|node|3|node|3|node}}
!colspan=2|Folding||Degree||Coxeter Plane
!colspan=2|Folding||Degree||Coxeter Plane
|- align=center
|-
|{{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c2}} || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|{{CDD|node_c2|3|node_c2|3|node_c3}} || {{CDD|node_c2|3|node_c3}} ||3||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c1}} || {{CDD|node_c1}} ||2||A<sub>1</sub>
|{{CDD|node_c2|3|node_c2|3|node_c2}} || {{CDD|node_c2}} ||2||A<sub>1</sub>
|}
|}


Line 93: Line 121:
|+ Example: B<sub>3</sub>, {{CDD|node|4|node|3|node}}
|+ Example: B<sub>3</sub>, {{CDD|node|4|node|3|node}}
!colspan=2|Folding||Degree||Coxeter Plane
!colspan=2|Folding||Degree||Coxeter Plane
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|4|node_c2|3|node_c1}} || {{CDD|node_c1|6|node_c2}} ||6||B<sub>3</sub>
|{{CDD|node_c2|4|node_c3|3|node_c2}} || {{CDD|node_c2|6|node_c3}} ||6||B<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|4|node_c2|3|node_c2}} || {{CDD|node_c1|4|node_c2}} ||4||B<sub>2</sub>
|{{CDD|node_c2|4|node_c2|3|node_c3}} || {{CDD|node_c2|6|node_c3}} ||3×2||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|4|node_c1|3|node_c2}} || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|{{CDD|node_c2|4|node_c3|3|node_c3}} || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|4|node_c1|3|node_c1}} || {{CDD|node_c1}} ||2||A<sub>1</sub>
|{{CDD|node_c2|4|node_c2|3|node_c2}} || {{CDD|node_c2}} ||2||A<sub>1</sub>
|}
|}


Line 107: Line 135:
|+ Example: H<sub>3</sub>, {{CDD|node|5|node|3|node}}
|+ Example: H<sub>3</sub>, {{CDD|node|5|node|3|node}}
!colspan=2|Folding||Degree||Coxeter Plane
!colspan=2|Folding||Degree||Coxeter Plane
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|5|node_c2|3|node_c1}} || {{CDD|node_c1|10|node_c2}} ||6||H<sub>3</sub>
|{{CDD|node_c2|5|node_c3|3|node_c2}} || {{CDD|node_c2|10|node_c3}} ||10||H<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|5|node_c2|3|node_c2}} || {{CDD|node_c1|5|node_c2}} ||5||H<sub>2</sub>
|{{CDD|node_c2|5|node_c3|3|node_c3}} || {{CDD|node_c2|10|node_c3}} ||5×2||H<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|5|node_c1|3|node_c2}} || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|{{CDD|node_c2|5|node_c2|3|node_c3}} || {{CDD|node_c2|3|node_c3}} ||3×2||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|5|node_c1|3|node_c1}} || {{CDD|node_c1}} ||2||A<sub>1</sub>
|{{CDD|node_c2|5|node_c2|3|node_c2}} || {{CDD|node_c2}} ||2||A<sub>1</sub>
|}
|}


Line 122: Line 150:
|+ Example: A<sub>4</sub>, {{CDD|node|3|node|3|node|3|node}}
|+ Example: A<sub>4</sub>, {{CDD|node|3|node|3|node|3|node}}
!colspan=3|Folding||Degree||Coxeter Plane
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|-
|{{CDD|node_c1|3|node_c2|3|node_c1|3|node_c2}}|| || {{CDD|node_c1|5|node_c2}} ||5||A<sub>4</sub>
|{{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3}}|| || {{CDD|node_c2|5|node_c3}} ||5||A<sub>4</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1}} || {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c3|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|3|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c2|3|node_c3|3|node_c3}}<BR>{{CDD|node_c2|3|node_c2|3|node_c2|3|node_c3}} || || {{CDD|node_c2|3|node_c3}} ||3||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c2|3|node_c2}} || || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|{{CDD|node_c2|3|node_c2|3|node_c2|3|node_c2}} || || {{CDD|node_c2}} || 2||A<sub>1</sub>
|-
|}

|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c2}} || || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
==B4==
|-

|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1}} || || {{CDD|node_c1}} || 2||A<sub>1</sub>
{| class=wikitable
|+ Example: B<sub>4</sub>, {{CDD|node|4|node|3|node|3|node}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|{{CDD|node_c2|4|node_c3|3|node_c2|3|node_c3}}|| || {{CDD|node_c2|8|node_c3}} ||8||B<sub>4</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|4|node_c3|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|4|node_c3|3|node_c2|3|node_c2}} || {{CDD|node_c2|4|node_c3|3|node_c2}} || {{CDD|node_c2|6|node_c3}} ||6||B<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c3}}<BR>{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2||A<sub>2</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c3|3|node_c3|3|node_c3}} || || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub>
|- align=center
|{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c2}} || || {{CDD|node_c2}} || 2||A<sub>1</sub>
|}

==D4==

{| class=wikitable
|+ Example: D<sub>4</sub>, {{CDD|node|3|node|split1|nodes}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|3|node_c3|split1|nodeab_c2}} ||{{CDD|node_c2|3|node_c3|4|node_c2}} ||{{CDD|node_c2|6|node_c3}} || 6||D<sub>4</sub>=B<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|3|node_c3|split1|nodeab_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2 ||A<sub>2</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{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>
|- align=center BGCOLOR="#ffe0e0"
| || || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub>
|- align=center
|{{CDD|node_c2|3|node_c2|split1|nodeab_c2}} || || {{CDD|node_c2}} ||2 ||A<sub>1</sub>
|}
|}


Line 141: Line 201:
|+ Example: F<sub>4</sub>, {{CDD|node|3|node|4|node|3|node}}
|+ Example: F<sub>4</sub>, {{CDD|node|3|node|4|node|3|node}}
!colspan=3|Folding||Degree||Coxeter Plane
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|-
|{{CDD|node_c1|3|node_c2|4|node_c1|3|node_c2}} || || {{CDD|node_c1|12|node_c2}} ||12||F<sub>4</sub>
|{{CDD|node_c2|3|node_c3|4|node_c2|3|node_c3}} || || {{CDD|node_c2|12|node_c3}} ||12||F<sub>4</sub>
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}} || {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c3|4|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|8|node_c3}} ||4×2||A<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|-
|{{CDD|node_c1|3|node_c1|4|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2|3|node_c1}} || {{CDD|node_c1|6|node_c2}} ||6||B<sub>3</sub>
|{{CDD|node_c2|3|node_c2|4|node_c3|3|node_c3}} || || {{CDD|node_c2|8|node_c3}} ||4×2||B<sub>2</sub>
|- align=center BGCOLOR="#e0e0ff"
|-
|{{CDD|node_c1|3|node_c1|4|node_c2|3|node_c2}} || || {{CDD|node_c1|4|node_c2}} ||4||B<sub>2</sub>
|{{CDD|node_c2|3|node_c2|4|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3|3|node_c2}} || {{CDD|node_c2|6|node_c3}} ||6||B<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|-
|{{CDD|node_c1|3|node_c1|4|node_c1|3|node_c2}} || || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|{{CDD|node_c2|3|node_c2|4|node_c2|3|node_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|4|node_c1|3|node_c1}} || || {{CDD|node_c1}} || 2||A<sub>1</sub>
|{{CDD|node_c2|3|node_c2|4|node_c2|3|node_c2}} || || {{CDD|node_c2}} || 2||A<sub>1</sub>
|}

==H4==

{| class=wikitable
|+ Example: H<sub>4</sub>, {{CDD|node|5|node|3|node|3|node}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|{{CDD|node_c2|5|node_c3|3|node_c2|3|node_c3}}|| || {{CDD|node_c2|3x|0x|node_c3}} ||30||H<sub>4</sub>
|- align=center
| || ||{{CDD|node_c2|20|node_c3}} ||20 ||
|- align=center
| || ||{{CDD|node_c2|12|node_c3}} ||12 || F<sub>4</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|5|node_c3|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|5|node_c3|3|node_c2|3|node_c2}} || {{CDD|node_c2|5|node_c3|3|node_c2}} || {{CDD|node_c2|10|node_c3}} ||10||H<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|5|node_c3|3|node_c3|3|node_c3}} || || {{CDD|node_c2|10|node_c3}} ||5×2||H<sub>2</sub>
|- align=center
|{{CDD|node_c2|5|node_c2|3|node_c3|3|node_c3}}<BR>{{CDD|node_c2|5|node_c2|3|node_c2|3|node_c3}} || || {{CDD|node_c2|3|node_c3}} ||3×2||A<sub>2</sub>
|- align=center
|{{CDD|node_c2|5|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center
|{{CDD|node_c2|5|node_c2|3|node_c2|3|node_c2}} || || {{CDD|node_c2}} || 2||A<sub>1</sub>
|}
|}


Line 159: Line 242:
{| class=wikitable
{| class=wikitable
|+ Example: A<sub>5</sub>, {{CDD|node|3|node|3|node|3|node|3|node}}
|+ Example: A<sub>5</sub>, {{CDD|node|3|node|3|node|3|node|3|node}}
!colspan=2|Folding||Degree||Coxeter Plane
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|-
|{{CDD|node_c1|3|node_c2|3|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|6|node_c2}} ||6||A<sub>5</sub>
|{{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3|3|node_c2}} || || {{CDD|node_c2|6|node_c3}} ||6||A<sub>5</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c2|3|node_c1|3|node_c1|3|node_c2}} = {{CDD|node_c1|3|node_c2|3|node_c1|3|node_c2}} || {{CDD|node_c1|5|node_c2}} ||5||A<sub>4</sub>
|{{CDD|node_c2|3|node_c2|3|node_c3|3|node_c2|3|node_c3}}<BR>{{CDD|node_c2|3|node_c3|3|node_c2|3|node_c2|3|node_c3}} || {{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3}} || {{CDD|node_c2|5|node_c3}} ||5||A<sub>4</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c2|3|node_c1|3|node_c2}} = {{CDD|node_c1|3|node_c2|3|node_c1|3|node_c2}} || {{CDD|node_c1|5|node_c2}} ||5||A<sub>4</sub>
|{{CDD|node_c2|3|node_c2|3|node_c2|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|3|node_c2|3|node_c3|3|node_c2|3|node_c2}}<BR>{{CDD|node_c2|3|node_c2|3|node_c3|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|3|node_c3|3|node_c3|3|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c2|3|node_c1|3|node_c1}} = {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c2|3|node_c2|3|node_c2|3|node_c3}} || || {{CDD|node_c2|3|node_c3}} ||3||A<sub>2</sub>
|- align=center
|-
|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c2|3|node_c1}} = {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|{{CDD|node_c2|3|node_c2|3|node_c2|3|node_c2|3|node_c2}} || || {{CDD|node_c2}} ||2||A<sub>1</sub>
|-
|}
|{{CDD|node_c1|3|node_c2|3|node_c2|3|node_c1|3|node_c1}} = {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|-
|{{CDD|node_c1|3|node_c2|3|node_c2|3|node_c2|3|node_c1}} = {{CDD|node_c1|3|node_c2|3|node_c1}} || {{CDD|node_c1|4|node_c2}} ||4||A<sub>3</sub>
|-
|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c2|3|node_c2}} || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|-
|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1|3|node_c2}} || {{CDD|node_c1|3|node_c2}} ||3||A<sub>2</sub>
|-
|{{CDD|node_c1|3|node_c1|3|node_c1|3|node_c1|3|node_c1}} || {{CDD|node_c1}} ||2||A<sub>1</sub>


==B5==

{| class=wikitable
|+ Example: B<sub>5</sub>, {{CDD|node|4|node|3|node|3|node|3|node}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c3|3|node_c2|3|node_c3|3|node_c2}} || || {{CDD|node_c2|10|node_c3}} ||10||B<sub>5</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c2|3|node_c3}} || {{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3}} || {{CDD|node_c2|10|node_c3}} ||5×2||A<sub>4</sub>
|- align=center
|{{CDD|node_c2|4|node_c3|3|node_c2|3|node_c2|3|node_c3}} || {{CDD|node_c2|4|node_c3|3|node_c2|3|node_c3}} || {{CDD|node_c2|8|node_c3}} ||8||B<sub>4</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|4|node_c3|3|node_c3|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|4|node_c3|3|node_c3|3|node_c2|3|node_c2}}<BR>{{CDD|node_c2|4|node_c3|3|node_c2|3|node_c2|3|node_c2}} || {{CDD|node_c2|4|node_c3|3|node_c2}} || {{CDD|node_c2|6|node_c3}} ||6||B<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c2|3|node_c3}}<BR>{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c3|3|node_c3}}<BR>{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c3|3|node_c3}} || || {{CDD|node_c2|6|node_c3}} ||3×2||A<sub>2</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c3|3|node_c2}}<BR>{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c2|3|node_c2}}<BR>{{CDD|node_c2|4|node_c2|3|node_c3|3|node_c3|3|node_c2}} || {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||A<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|4|node_c3|3|node_c3|3|node_c3|3|node_c3}} || || {{CDD|node_c2|4|node_c3}} ||4||B<sub>2</sub>
|- align=center
|{{CDD|node_c2|4|node_c2|3|node_c2|3|node_c2|3|node_c2}} || || {{CDD|node_c2}} ||2||A<sub>1</sub>
|}

==D5==

{| class=wikitable
|+ Example: D<sub>5</sub>, {{CDD|node|3|node|3|node|split1|nodes}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|{{CDD|node_c2|3|node_c3|3|node_c2|split1|nodeab_c3}} ||{{CDD|node_c2|3|node_c3|3|node_c2|4|node_c3}} ||{{CDD|node_c2|8|node_c3}} || 8||D<sub>5</sub>=B<sub>4</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|3|node_c2|3|node_c3|split1|nodeab_c2}}<BR>{{CDD|node_c2|3|node_c3|3|node_c3|split1|nodeab_c2}} ||{{CDD|node_c2|3|node_c3|split1|nodeab_c2}} = {{CDD|node_c2|3|node_c3|4|node_c2}} || {{CDD|node_c2|6|node_c3}} ||6 ||D<sub>4</sub>=B<sub>3</sub>
|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|3|node_c3|3|node_c3|split1|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c2|3|node_c3|split1|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c2|3|node_c2|split1|nodeab_c2-3}} || || {{CDD|node_c2|6|node_c3}} ||3×2||A<sub>2</sub>
|- align=center
|{{CDD|node_c2|3|node_c3|3|node_c2|split1|nodeab_c2-3}} ||{{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3}} || {{CDD|node_c2|5|node_c3}} ||5||A<sub>4</sub>
|- align=center
|{{CDD|node_c2|3|node_c3|3|node_c2|split1|nodeab_c2}}<BR>{{CDD|node_c2|3|node_c2|3|node_c2|split1|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c3|3|node_c3|split1|nodeab_c2-3}}<BR>{{CDD|node_c2|3|node_c2|3|node_c3|split1|nodeab_c2-3}}
||{{CDD|node_c3|split1|nodeab_c2}} = {{CDD|node_c2|3|node_c3|3|node_c2}} || {{CDD|node_c2|4|node_c3}} ||4||D<sub>3</sub>=A<sub>3</sub>
|- align=center
|{{CDD|node_c2|3|node_c2|3|node_c2|split1|nodeab_c2}} || || {{CDD|node_c2}} ||2 ||A<sub>1</sub>
|}

==E6==

{| class=wikitable
|+ Example: E<sub>6</sub>, {{CDD|node|3|node|split1|nodes|3ab|nodes}}
!colspan=3|Folding||Degree||Coxeter Plane
|- align=center
|{{CDD|node_c2|3|node_c3|split1|nodeab_c2|3ab|nodeab_c3}} || {{CDD|node_c2|3|node_c3|4|node_c2|3|node_c3}} ||{{CDD|node_c2|12|node_c3}} || 12||E<sub>6</sub> = F<sub>4</sub>
|- align=center
| || ||{{CDD|node_c2|9|node_c3}} || 9||

|- align=center BGCOLOR="#ffe0e0"
|{{CDD|node_c2|3|node_c3|split1|nodeab_c2-3|3ab|nodeab_c3-2}} || {{CDD|node_c3|3|node_c2|3|node_c3|split1|nodeab_c2}} = {{CDD|node_c3|3|node_c2|3|node_c3|4|node_c2}} ||{{CDD|node_c2|8|node_c3}} || 8||D<sub>5</sub> = B<sub>4</sub>

|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|3|node_c2|split1|nodeab_c3|3ab|nodeab_c2}} || {{CDD|node_c2|split1|nodeab_c3|3ab|nodeab_c2}} ||{{CDD|node_c2|6|node_c3}} || 6||A<sub>5</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|3|node_c3|split1|nodeab_c2|3ab|nodeab_c2}}<BR>{{CDD|node_c2|3|node_c3|split1|nodeab_c3-2|3ab|nodeab_c2}}<BR>{{CDD|node_c2|3|node_c3|split1|nodeab_c2|3ab|nodeab_c3-2}}<BR>{{CDD|node_c2|3|node_c3|split1|nodeab_c2|3ab|nodeab_c3}}
|| {{CDD|node_c2|3|node_c3|split1|nodeab_c2}} = {{CDD|node_c2|3|node_c3|4|node_c2}} ||{{CDD|node_c2|6|node_c3}} || 6||D<sub>4</sub> = B<sub>3</sub>
|- align=center BGCOLOR="#e0e0ff"
|{{CDD|node_c2|3|node_c3|split1|nodeab_c3|3ab|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c2|3ab|nodeab_c3-2}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c3-2|3ab|nodeab_c3-2}}|| || {{CDD|node_c2|6|node_c3}} || 3×2||A<sub>2</sub>
|- align=center
|{{CDD|node_c2|3|node_c3|split1|nodeab_c3-2|3ab|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c3|3ab|nodeab_c3-2}} <BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c3-2|3ab|nodeab_c2-3}}
||{{CDD|node_c2|3|node_c3|3|node_c2|3|node_c3}} ||{{CDD|node_c2|5|node_c3}} || 5||A<sub>4</sub>
|- align=center
|{{CDD|node_c2|3|node_c2|split1|nodeab_c3-2|3ab|nodeab_c2}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c3|3ab|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c3|split1|nodeab_c3|3ab|nodeab_c3-2}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c2|3ab|nodeab_c3}}<BR>{{CDD|node_c2|3|node_c2|split1|nodeab_c2-3|3ab|nodeab_c3}}<BR>{{CDD|node_c3|3|node_c2|split1|nodeab_c3-2|3ab|nodeab_c3-2}}
|| {{CDD|node_c2|split1|nodeab_c3}} || {{CDD|node_c2|4|node_c3}} || 4||A<sub>3</sub>
|- align=center
|{{CDD|node_c2|3|node_c2|split1|nodeab_c2|3ab|nodeab_c2}}|| || {{CDD|node_c2}} || 2||A<sub>1</sub>
|}
|}

Latest revision as of 02:20, 31 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 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
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