Embedding of a Simple Lie Group into a Simple Lie Group and Branching Rules

A criterion established by Dynkin is used to specify the embedding of a connected simple Lie group $\mathcal{G}\prime$ into a connected simple Lie group $\mathcal{G}$ , and to derive a standard procedure for evaluating branching rules. It is shown that the weight systems of the irreducible parts contained in the representation of $\mathcal{G}\prime$ induced by a given finite dimensional representation φ of $\mathcal{G}$ are obtained by projection of the weight system of φ. The projection mapping is determined directly from the specification of the embedding. The general procedure is supplemented with two constraint equations on the dimensions and indices of the irreducible representations.