When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfulling this is introduced for the tensor product of three irreps of su(2). This method is then extended to the tensor product of two irreps of su(3). In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over C .
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