The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.