Template:Dynkin/testcases

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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> 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> 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> 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) 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 corresponds 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> 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> 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> 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) 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 corresponds 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.