Difference between revisions of "Template:Dynkin/testcases"

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imported>JohnBlackburne
(→‎Testing sandbox version: Add table of examples, using /sandbox version)
imported>JohnBlackburne
(→‎Testing main template: same examples using normal template)
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{{tlx|{{TEMPLATENAMEE|require=testcases}}}}
 
{{tlx|{{TEMPLATENAMEE|require=testcases}}}}
  
{{ {{TEMPLATENAMEE|require=testcases}} }}
+
 
 +
 
 +
{| class=wikitable width=640
 +
|+ Rank 2 Dynkin diagrams
 +
|-
 +
!rowspan=2|Group<BR>name
 +
!colspan=2|Dynkin diagram
 +
!colspan=2|[[Cartan matrix]]
 +
!rowspan=2|Symmetry<BR>order
 +
!rowspan=2|Related<BR>simply-laced<BR>automorphic<BR>group<sup>3</sup>
 +
|-
 +
!(Standard)<BR>multi-edged<BR>graph<sup>1</sup>
 +
!Valued<BR>graph<sup>2</sup>
 +
!<math>\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ]</math>
 +
![[Determinant]]<P>(4-a<sub>21</sub>*a<sub>12</sub>)
 +
|- align=center
 +
!colspan=7|[[Dynkin_diagram#Finite_Dynkin_diagrams|Finite]] (Determinant>0)
 +
|- align=center
 +
!A<sub>1</sub>xA<sub>1</sub>
 +
|{{Dynkin|node}} {{Dynkin|node}}
 +
|{{Dynkin|node}} {{Dynkin|node}}
 +
|<math>\left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ]</math>
 +
|4
 +
|2
 +
|&nbsp;
 +
|- align=center
 +
!A<sub>2</sub>
 +
|{{Dynkin|node|3|node}}
 +
|{{Dynkin|node|3|node}}
 +
|<math>\left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ]</math>
 +
|3
 +
|3
 +
|&nbsp;
 +
|- align=center
 +
!B<sub>2</sub>
 +
|{{Dynkin|node|4b|nodeg}}
 +
|{{Dynkin|node|v12|nodeg}}
 +
|<math>\left [\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}\right ]</math>
 +
|2
 +
|4
 +
|<math>{A}_3</math> {{Dynkin|node|branch2}}
 +
|- align=center
 +
!C<sub>2</sub>
 +
|{{Dynkin|nodeg|4a|node}}
 +
|{{Dynkin|nodeg|v21|node}}
 +
|<math>\left [\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}\right ]</math>
 +
|2
 +
|4
 +
|<math>{A}_3</math> {{Dynkin|branch1|node}}
 +
|- align=center
 +
!G<sub>2</sub>
 +
|{{Dynkin|nodeg|6a|node}}
 +
|{{Dynkin|nodeg|v31|node}}
 +
|<math>\left [\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}\right ]</math>
 +
|1
 +
|6
 +
|<math>{D}_4</math> [[File:Dynkin_affine_D3_folding.png]]
 +
|- align=center
 +
!colspan=7|[[Dynkin_diagram#Affine_Dynkin_diagrams|Affine]] (Determinant=0)
 +
|- align=center
 +
!A<sub>1</sub><sup>(1)</sup>
 +
|{{Dynkin|nodeg|4ab|nodeg}}
 +
|{{Dynkin|nodeg|v22|nodeg}}
 +
|<math>\left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ]</math>
 +
|0
 +
|&infin;
 +
|<math>{\tilde{A}}_3</math> [[File:Dynkin_affine_A3_folding.png]]
 +
|- align=center
 +
!A<sub>2</sub><sup>(2)</sup>
 +
|{{Dynkin|nodeg|4c|node}}
 +
|{{Dynkin|nodeg|v41|node}}
 +
|<math>\left [\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}\right ]</math>
 +
|0
 +
|&infin;
 +
|<math>{\tilde{D}}_4</math> [[File:Dynkin_affine_D4_folding.png]]
 +
|- align=center
 +
!colspan=7|Hyperbolic (Determinant<0)
 +
|- align=center
 +
!
 +
|colspan=2|{{Dynkin|nodeg|v51|node}}
 +
|<math>\left [\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}\right ]</math>
 +
| -1
 +
|&infin;
 +
|H<sub>5</sub><sup>(6)</sup> [[File:Dynkin_hyperbolic_pentstar_folding.png]]
 +
|- align=center
 +
!
 +
|colspan=2|{{Dynkin|nodeg|vab|nodeg}}
 +
|<math>\left [\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}\right ]</math>
 +
|4-ab
 +
|&infin;
 +
|-
 +
|colspan=7|
 +
Note<sup>1</sup>: The multi-edged diagram correponds to the nondiagonal Cartan matrix elements a<sub>21</sub>, a<sub>12</sub>, with the number of edges drawn equal to max(a<sub>21</sub>, a<sub>12</sub>), and an arrow pointing towards nonunity element(s).
 +
 
 +
Note<sup>2</sup>: For hyperbolic groups, (a<sub>12</sub>*a<sub>21</sub>>4), the multiedge style is abandoned in favor of an explicit labeling (a<sub>21</sub>, a<sub>12</sub>) on the edge. These are usually not applied to finite and affine graphs.
 +
 
 +
Note<sup>3</sup>: Many multi-edged groups are [[automorphism|automorphic]] via a [[Dynkin_diagram#Folding|folding operation]] with a higher ranked simply-laced group.
 +
|}

Revision as of 00:43, 23 April 2014

Testing sandbox version

{{Dynkin/sandbox}}


Rank 2 Dynkin diagrams
Group
name
Dynkin diagram Cartan matrix Symmetry
order
Related
simply-laced
automorphic
group3
(Standard)
multi-edged
graph1
Valued
graph2
<math>\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ]</math> Determinant

(4-a21*a12)

Finite (Determinant>0)
A1xA1 Script error: No such module "Dynkin". Script error: No such module "Dynkin". Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ]</math> 4 2  
A2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ]</math> 3 3  
B2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}\right ]</math> 2 4 <math>{A}_3</math> Script error: No such module "Dynkin".
C2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}\right ]</math> 2 4 <math>{A}_3</math> Script error: No such module "Dynkin".
G2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}\right ]</math> 1 6 <math>{D}_4</math> File:Dynkin affine D3 folding.png
Affine (Determinant=0)
A1(1) Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ]</math> 0 <math>{\tilde{A}}_3</math> File:Dynkin affine A3 folding.png
A2(2) Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}\right ]</math> 0 <math>{\tilde{D}}_4</math> File:Dynkin affine D4 folding.png
Hyperbolic (Determinant<0)
Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}\right ]</math> -1 H5(6) File:Dynkin hyperbolic pentstar folding.png
Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}\right ]</math> 4-ab

Note1: The multi-edged diagram correponds to the nondiagonal Cartan matrix elements a21, a12, with the number of edges drawn equal to max(a21, a12), and an arrow pointing towards nonunity element(s).

Note2: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs.

Note3: Many multi-edged groups are automorphic via a folding operation with a higher ranked simply-laced group.


Testing main template

{{Dynkin}}


Rank 2 Dynkin diagrams
Group
name
Dynkin diagram Cartan matrix Symmetry
order
Related
simply-laced
automorphic
group3
(Standard)
multi-edged
graph1
Valued
graph2
<math>\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ]</math> Determinant

(4-a21*a12)

Finite (Determinant>0)
A1xA1 Script error: No such module "Dynkin". Script error: No such module "Dynkin". Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ]</math> 4 2  
A2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ]</math> 3 3  
B2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}\right ]</math> 2 4 <math>{A}_3</math> Script error: No such module "Dynkin".
C2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}\right ]</math> 2 4 <math>{A}_3</math> Script error: No such module "Dynkin".
G2 Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}\right ]</math> 1 6 <math>{D}_4</math> File:Dynkin affine D3 folding.png
Affine (Determinant=0)
A1(1) Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ]</math> 0 <math>{\tilde{A}}_3</math> File:Dynkin affine A3 folding.png
A2(2) Script error: No such module "Dynkin". Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}\right ]</math> 0 <math>{\tilde{D}}_4</math> File:Dynkin affine D4 folding.png
Hyperbolic (Determinant<0)
Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}\right ]</math> -1 H5(6) File:Dynkin hyperbolic pentstar folding.png
Script error: No such module "Dynkin". <math>\left [\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}\right ]</math> 4-ab

Note1: The multi-edged diagram correponds to the nondiagonal Cartan matrix elements a21, a12, with the number of edges drawn equal to max(a21, a12), and an arrow pointing towards nonunity element(s).

Note2: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs.

Note3: Many multi-edged groups are automorphic via a folding operation with a higher ranked simply-laced group.