Root Systems

Choose a type and rank, and a labelling of the Dynkin diagram, then click View: you get the Dynkin diagram, the Cartan matrix, the fundamental weights and coweights, ρ, the order of the Weyl group, the list of positive roots, and the matrices relating the standard and weight coordinates. Download saves the same data as a text file.

Types F* and G* are the root systems 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

        2
        |
        |
1---3---4---5---6---7---8

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

        8
        |
        |
1---2---3---4---5---6---7