Weight Diagrams of Weight Two

This file contains a Mathematica program to display weight diagrams for irreducible representations of the three simply connected rank two semisimple complex Lie groups, of type A2 (i.e. SL(3,C)), B2 (i.e. Sp(2,C)) and G2 (the exceptional group).

The weight diagram of a representation R describes the restriction of the representation to the maximal Cartan subgroup H. The irreducible representations of the commutative group H are one-dimensional, and may be parametrized by vectors in Euclidean space---two dimensional Euclidean space for the three rank two Lie groups. These vectors (called weights) are the points in the diagram. It is natural to label them with an integer which is the multiplicity of the weight. Thus the sum of the multiplicities is the dimension of the representation R.

There is a natural partial ordering of the weights, and the representation R is naturally parametrized by its highest weight vector.

Usage of the program is simple. After loading the file,

WeightDiagramA2[5,4]

will produce this output:

This is the weight diagram of the irreducible representation of SL(3,C) having highest weight vector (5,4).

WeightDiagramB2[2,5]

will produce this output:

This is the weight diagram of the irreducible representation of Sp(4,C) having highest weight vector (2,5). And:

WeightDiagramG2[2,3]

will produce this output:

This is the weight diagram of the irreducible representation of the exceptional group G2 having highest weight vector (2,3).

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Last modified: Sun Dec 22 08:11:19 PST 1996