Template:Dynkin/testcases

Testing sandbox version

Rank 2 Dynkin diagrams

Group name	Dynkin diagram		Cartan matrix		Symmetry order	Related simply-laced automorphic group3
	(Standard) multi-edged graph1	Valued graph2	$\left[\begin{array}{cc}2& a_{12}\\ a_{21}& 2\end{array}\right]$	Determinant (4-a21*a12)
Finite (Determinant>0)
A1xA1	Template:Dynkin/sandbox Template:Dynkin/sandbox	Template:Dynkin/sandbox Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$	4	2
A2	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$	3	3
B2	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -2\\ -1& 2\end{array}\right]$	2	4	$A_3$ Template:Dynkin
C2	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -1\\ -2& 2\end{array}\right]$	2	4	$A_3$ Template:Dynkin
G2	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$	1	6	$D_4$
Affine (Determinant=0)
A1(1)	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$	0	∞	${\tilde{A}}_3$
A2(2)	Template:Dynkin/sandbox	Template:Dynkin/sandbox	$\left[\begin{array}{cc}2& -1\\ -4& 2\end{array}\right]$	0	∞	${\tilde{D}}_4$
Hyperbolic (Determinant<0)
	Template:Dynkin/sandbox		$\left[\begin{array}{cc}2& -1\\ -5& 2\end{array}\right]$	-1	∞	H5(6)
	Template:Dynkin/sandbox		$\left[\begin{array}{cc}2& -b\\ -a& 2\end{array}\right]$	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.

Template:Clear

Testing main template

Rank 2 Dynkin diagrams

Group name	Dynkin diagram		Cartan matrix		Symmetry order	Related simply-laced automorphic group3
	(Standard) multi-edged graph1	Valued graph2	$\left[\begin{array}{cc}2& a_{12}\\ a_{21}& 2\end{array}\right]$	Determinant (4-a21*a12)
Finite (Determinant>0)
A1xA1	Template:Dynkin Template:Dynkin	Template:Dynkin Template:Dynkin	$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$	4	2
A2	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$	3	3
B2	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -2\\ -1& 2\end{array}\right]$	2	4	$A_3$ Template:Dynkin
C2	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -1\\ -2& 2\end{array}\right]$	2	4	$A_3$ Template:Dynkin
G2	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$	1	6	$D_4$
Affine (Determinant=0)
A1(1)	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]$	0	∞	${\tilde{A}}_3$
A2(2)	Template:Dynkin	Template:Dynkin	$\left[\begin{array}{cc}2& -1\\ -4& 2\end{array}\right]$	0	∞	${\tilde{D}}_4$
Hyperbolic (Determinant<0)
	Template:Dynkin		$\left[\begin{array}{cc}2& -1\\ -5& 2\end{array}\right]$	-1	∞	H5(6)
	Template:Dynkin		$\left[\begin{array}{cc}2& -b\\ -a& 2\end{array}\right]$	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.