Root Systems

This page contains information about root systems, including choices of coordinates.

It is useful in conjuction with the Spherical Unitary Explorer

Type: Rank:

Labelling of the Dynkin Diagram:Other:

Types F* and G* are the roots sytems of type F and G, with coordinates dual to the standard (Bourbaki) coordinates.

There are two standard labellings of the Dynkin Diagram: Bourbaki and Gap. These differ in types B,C,D and G. In the Bourbaki labelling, in types B,C, and D, the double bond or fork is near the node labelled n. The Gap labelling is the opposite of this. (In G2 the short root is labelled 1 in the Bourbaki labelling, and 2 in Gap.) The Gap labelling is useful because the root system spanned by roots 1,2,...,k is of the same type.

For example in type B3:


It is also possible to choose your own labelling of the Dynkin Diagram. The matrices will be modified accordingly. Enter a permutation of 1,2,...,n where n is the rank. These give the numbering of the nodes with respect to the Bourbaki numbering. For example in E8 the Bourbaki numbering is

so if you enter 1 8 2 3 4 5 6 7 for Other you will get