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Abstract: This paper focuses on the tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary irreducible representations of . We will describe an explicit basis for this algebra. This construction relates directly with the combinatorial description of Littlewood–Richardson coefficients in terms of Littlewood–Richardson tableaux. Philosophically, one may view this construction as a recasting of the Littlewood–Richardson rule in the context of classical invariant theory. [Copyright &y& Elsevier]