The place of the adjoint representation in the Kronecker square of irreducible representations of simple Lie groups

The multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite-dimensional irreducible representation lambda of any compact simple Lie group is shown to be equal to the number of non-vanishing components of the Dynkin label of lambda. The resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of lambda is discussed, with complete results being found for all of the classical and some of the exceptional simple Lie groups, and partial results culminating in conjectures for the remaining exceptional groups.