Semisimple Subgroups of Semisimple Groups

Abstract

The projection theorem for weights of a representation of a semisimple group G on restriction to a semisimple subgroup is derived, and the existence of a subgroup corresponding to a given projection is discussed. Dynkin's definition of the index of a simple subgroup is extended to the case of G being only semisimple, and the geometrical meaning of the index is given. A method is developed for finding branching rules for both regular and nonregular subgroups. Explicit general formulas for the branching multiplicities are obtained for all cases when G is of rank 2 and for B₃(R₇) → G₂. Applications to the construction of weight diagrams and the ``state‐labeling'' problems for B₂ and G₂ are mentioned.