The state labeling problem—A universal solution

Abstract

A new method is proposed for labeling a basis for any irreducible representation, with top weight, of a simple Lie algebra (LA). It has two advantages over Gel’fand–Zetlin patterns: (i) being exactly the same for all LA’s both exceptional and classical and (ii) providing labels that are much more compact. Thus it may prove useful in discussing large representations such as those E8⊗E8 that occur in superstring theory. The method makes essential use of the Weyl group and is based on a theorem that associates to each weight a symmetric matrix with integer coefficients whose rank equals the multiplicity of the weight.