Details for Cartan matrices and Dynkin diagrams
|
Description
|
|
•
|
This document contains a list of all Cartan matrices and Dynkin diagrams for all classical root types of rank and for all exceptional root types.
|
|
|
Code
|
|
>
|
with(DifferentialGeometry): with(LieAlgebras):
|
We give 3 simple programs. The first calculates the edge matrix and the second the relative lengths 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]);
|
>
|
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)
|
>
|
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);
|
|
|
Root Type A
|
|
Root Type
|
Cartan Matrix
|
|
|
Dynkin Diagram
|
|
|
--
|
--
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here is the Cartan matrix for and the corresponding edge matric and root length vector
>
|
C := CartanMatrix("A", 4);
|
| (3.1) |
>
|
E := EdgeMatrix(C); L := RootLengths(C);
|
| (3.2) |
Re -construct the Cartan matrix.
>
|
DynkinDiagramDataToCartanMatrix(E, L);
|
| (3.3) |
|
|
Root Type B
|
|
Root
Type
|
Cartan Matrix
|
|
|
Dynkin Diagram
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here is the Cartan matrix for and the corresponding edge matrix and root length vector.
>
|
C := CartanMatrix("B", 4);
|
| (4.1) |
>
|
E := EdgeMatrix(C); L := RootLengths(C);
|
| (4.2) |
Re -construct the Cartan matrix.
>
|
DynkinDiagramDataToCartanMatrix(E, L);
|
| (4.3) |
|
|
Root Type C
|
|
Root Type
|
Cartan Matrix
|
|
|
Dynkin Diagram
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here is the Cartan matrix for and the corresponding edge matrix and root length vector.
>
|
C := CartanMatrix("C", 4);
|
| (5.1) |
>
|
E := EdgeMatrix(C); L := RootLengths(C);
|
| (5.2) |
Re -construct the Cartan matrix.
>
|
DynkinDiagramDataToCartanMatrix(E, L);
|
| (5.3) |
|
|
Root Type D
|
|
Root
Type
|
Cartan Matrix
|
|
|
Dynkin Diagram
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here is the Cartan matrix for and the corresponding edge matrix and root length vector.
>
|
C := CartanMatrix("D", 4);
|
| (6.1) |
>
|
E := EdgeMatrix(C); L := RootLengths(C);
|
| (6.2) |
Re -construct the Cartan matrix.
>
|
DynkinDiagramDataToCartanMatrix(E, L);
|
| (6.3) |
|
|
Exceptional Root Types
|
|
Root Type
|
Cartan Matrix
|
|
|
Dynkin Diagram
|
|
|
|
--
|
|
|
|
|
--
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here is the Cartan matrix for and the corresponding edge matrix and root length vector.
>
|
C := CartanMatrix("E", 6);
|
| (7.1) |
>
|
E := EdgeMatrix(C); L := RootLengths(C);
|
| (7.2) |
Re -construct the Cartan matrix.
>
|
DynkinDiagramDataToCartanMatrix(E, L);
|
| (7.3) |
|
|