Inner and Restriction Multiplicity for Classical Groups

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Abstract
For the classical compact Lie groups G, a formula for the multiplicity of weights (called inner multiplicity) is given. This formula relates the inner multiplicity of a group G to the inner multiplicity of a naturally embedded subgroup G′. For the SU(n) groups the formula can be brought into a particularly simple form—namely, a sum over Kronecker symbols—by choosing the group SU(2) for G′. The multiplicity of irreducible representations of a subgroup G′ into which an irreducible representation of a group G decomposes if G is restricted to G′—called restriction multiplicity of G with respect to G′—is related to the inner multiplicity of the group G.