Cartan Matrices and Dynkin Diagram Details

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Details for Cartan matrices and Dynkin diagrams

Exceptional Root Types

Description

•	This document contains a list of all Cartan matrices and Dynkin diagrams for all classical root types of rank $\leq 6$ and for all exceptional root types.

•	Let $Δ_{0} = \{α_{1}, α_{2}, .. &period;, α_{m}\}$ be a set of simple roots and let $(\cdot, \cdot)$ be the inner product on the roots induced by the Killing form. The Cartan matrix is given by $C_{ij} = 2 \frac{(α_{i}, α_{j})}{(α_{j}, α_{j})} &period;$

•

From the Cartan matrix one can calculate the number of lines connecting $α_{i}$ to $α_{j}$ as $E_{ij} = C_{ij} C_{ji} (*)$ . The relative lengths of the root vectors can be found as the ratios $\frac{(α_{i}, α_{i})}{(α_{j}, α_{j})} = \frac{C_{ji}}{C_{ij}}$ (**). Set $L_{i} = \frac{(α_{i}, α_{i})}{(α_{i + 1}, α_{i + 1})} &period;$

•

The edge matrix $E_{ij}$ and the root length vector $L_{i}$ clearly determine the Dynkin diagram. Conversely, the equations (*) and (**), together with the facts that $C_{ii}$ = 2 and $- 3 \leq C_{ij} \leq 0$ for $i \neq j$ uniquely determine the Cartan matrix from the edge matrix $E_{ij}$ and the root length vector $L_{i} &period;$ For additional details see, for example, W. A. de Graaf, Lie Algebras: Theory and Algorithms, pages 167-168.

Code

>	with(DifferentialGeometry): with(LieAlgebras):

We give 3 simple programs. The first calculates the edge matrix $E_{ij}$ and the second the relative lengths $L_{i}$ of the roots. The third program re-constructs the Cartan matrix from the edge matrix and the relative length vector.

>	EdgeMatrix := proc(C) local n;

>	description `a procedure to find the adjacency matrix for the Dynkin diagram from the Cartan matrix`;

>	n := LinearAlgebra:-ColumnDimension(C);

>	Matrix(n, n, (i, j) -> C[i, j]*C[j, i]);

end:

>	RootLengths := proc(C) local n, Eq, soln;

>	description `a procedure to find the ratio of the root lengths for the Dynkin diagram from the Cartan matrix`;

>	n := LinearAlgebra:-ColumnDimension(C);

>	Eq := {seq(seq(C[j, i]*x\|\|i/x\|\|j = C[i, j], i = 1 .. n) ,j = 1 .. n)}:

>	soln := solve(Eq, {seq(x\|\|i , i = 1 .. n)});

>	eval(Vector([seq(x\|\|i/x\|\|(i+1), i = 1 .. n-1)]), soln)

end:

>	DynkinDiagramDataToCartanMatrix := proc(Edges, L) local n, C, vars, Eq1, Eq2, Eq3, soln;

>	description `a procedure to find the Cartan matrix from the Dynkin diagram (edge matrix and root length rations)`;

>	n := LinearAlgebra:-ColumnDimension(Edges);

>	C := Matrix(n, n, proc(i, j) if i=j then 2 else c\|\|i\|\|j fi end); vars := indets(C);

>	Eq1:= {seq(seq( C[i,j]*C[j,i] = Edges[i,j], j = i+1..n), i = 1..n)};

>	Eq2 := {seq(seq(C[j,i]*mul(L[k], k = i.. j-1) = C[i,j], j = i+1..n) , i = 1..n-1)};

>	Eq3 := {seq(v<=0 ,v =vars), seq(v >= -3, v=vars)};

>	soln := solve(Eq1 union Eq2 union Eq3, vars);

>	eval(C, [soln][1]);

end:

Root Type A

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$A_{1}$	$[\begin{array}{r} 2 \end{array}]$	--	--
$A_{2}$	$[\begin{array}{r} 2 & - 1 \\ - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 \\ 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \end{array}]$
$A_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \end{array}]$
$A_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 1 \\ 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}]$
$A_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
$A_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}]$

Here is the Cartan matrix for $A_{4}$ and the corresponding edge matric and root length vector

>	C := CartanMatrix("A", 4);

>	E := EdgeMatrix(C); L := RootLengths(C);

Re -construct the $A_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type B

Root Type	Cartan Matrix $C$	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$B_{2}$	$[\begin{array}{r} 2 & - 2 \\ - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 2 \\ 0 & 0 \end{array}]$	$[\begin{array}{r} 2 \end{array}]$
$B_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 2 \\ 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array}],$	$[\begin{array}{r} 1 \\ 2 \end{array}]$
$B_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 2 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]$
$B_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 2 \\ 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 2 \end{array}]$
$B_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 2 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}],$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 2 \end{array}]$

Here is the Cartan matrix for $B_{4}$ and the corresponding edge matrix and root length vector.

>	C := CartanMatrix("B", 4);

>	E := EdgeMatrix(C); L := RootLengths(C);

Re -construct the $B_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type C

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$C_{3}$	$[\begin{array}{r} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 \\ 1 & 4 & 2 \\ 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ \frac{1}{2} \end{array}]$
$C_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 2 \\ 0 & 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ 1 \\ \frac{1}{2} \end{array}]$
$C_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 2 \\ 0 & 0 & 0 & 2 & 4 \end{array}],$	$[\begin{array}{c} 1 \\ 1 \\ 1 \\ \frac{1}{2} \end{array}]$
$C_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 2 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 2 \\ 0 & 0 & 0 & 0 & 2 & 4 \end{array}]$	$[\begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \frac{1}{2} \end{array}]$

Here is the Cartan matrix for $C_{4}$ and the corresponding edge matrix and root length vector.

>	C := CartanMatrix("C", 4);

>	E := EdgeMatrix(C); L := RootLengths(C);

Re -construct the $C_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Root Type D

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$D_{3}$	$[\begin{array}{r} 2 & - 1 & - 1 \\ - 1 & 2 & 0 \\ - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 1 \\ 1 & 4 & 0 \\ 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \end{array}]$
$D_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & - 1 \\ 0 & - 1 & 2 & 0 \\ 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 1 \\ 0 & 1 & 4 & 0 \\ 0 & 1 & 0 & 4 \end{array}],$	$[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}]$
$D_{5}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & - 1 \\ 0 & 0 & - 1 & 2 & 0 \\ 0 & 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 1 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
$D_{6}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & - 1 \\ 0 & 0 & 0 & - 1 & 2 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 & 0 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 1 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}]$

Here is the Cartan matrix for $D_{4}$ and the corresponding edge matrix and root length vector.

>	C := CartanMatrix("D", 4);

>	E := EdgeMatrix(C); L := RootLengths(C);

Re -construct the $D_{4}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

Exceptional Root Types

Root Type	Cartan Matrix	$C_{ij} C_{ji}$	$\frac{C_{i, i + 1}}{C_{i + 1, i}}$	Dynkin Diagram
$E_{6}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	--
$E_{7}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$	--
$E_{8}$	$[\begin{array}{r} 2 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & - 1 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 2 & - 1 & 0 & 0 & 0 & 0 \\ 0 & - 1 & - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 4 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 4 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \end{array}]$
$F_{4}$	$[\begin{array}{r} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 2 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{array}]$	$[\begin{array}{r} 4 & 1 & 0 & 0 \\ 1 & 4 & 2 & 0 \\ 0 & 2 & 4 & 1 \\ 0 & 0 & 1 & 4 \end{array}]$	$[\begin{array}{r} 1 \\ 2 \\ 1 \end{array}]$
$G_{2}$	$[\begin{array}{r} 2 & - 1 \\ - 3 & 2 \end{array}]$	$[\begin{array}{r} 4 & 3 \\ 3 & 4 \end{array}] \begin{array}{r} \end{array}$	$[\begin{array}{c} \frac{1}{3} \end{array}]$

Here is the Cartan matrix for $E_{6}$ and the corresponding edge matrix and root length vector.

>	C := CartanMatrix("E", 6);

>	E := EdgeMatrix(C); L := RootLengths(C);

Re -construct the $E_{6}$ Cartan matrix.

>	DynkinDiagramDataToCartanMatrix(E, L);

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