Difference between revisions of "Template:Dynkin/testcases"
imported>JohnBlackburne (→Testing sandbox version: Add table of examples, using /sandbox version) |
imported>JohnBlackburne (→Testing main template: same examples using normal template) |
||
| Line 111: | Line 111: | ||
{{tlx|{{TEMPLATENAMEE|require=testcases}}}} | {{tlx|{{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 | ||
| + | | | ||
| + | |- 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 | ||
| + | | | ||
| + | |- 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 | ||
| + | |∞ | ||
| + | |<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 | ||
| + | |∞ | ||
| + | |<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 | ||
| + | |∞ | ||
| + | |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 | ||
| + | |∞ | ||
| + | |- | ||
| + | |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
| 42px | This is the template test cases page for the sandbox of Template:Dynkin. Script error: No such module "Purge". to update the examples. If there are many examples of a complicated template, later ones may break due to limits in MediaWiki, see the HTML comment "NewPP limit report" in the rendered page. You can also use Special:ExpandTemplates to examine the results of template uses. You can test how this page looks in the different skins with these links: |
Testing sandbox version
{{Dynkin/sandbox}}
| 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}}
| 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. | ||||||