Computers in Lie Algebras. II: Calculation of Outer Multiplicities

The tensor product of two irreducible representations $\rho _1 $ and $\rho _2 $ of a complex simple Lie algebra L is a completely reducible representation of L, but it is often not irreducible. Associated with each irreducible representation $\rho $ of L is its outer multiplicity with respect to $\rho _1 \otimes \rho _2 $. The outer multiplicity of $\rho $ is the number of times it appears in the decomposition of $\rho _1 \otimes \rho _2 $ into its sum of irreducible components. This paper starts with a survey of basic results about outer multiplicities and the structure of the tensor product of two representations. Then many of the methods of computing the outer multiplicities of a given tensor product are discussed. Finally computer implementations of the Racah formula and the method of successive subtractions are described. The Racah formula is very fast for small dimensioned representations and ranks less than 5. The method of successive subtractions will handle larger ranks and very large dimensioned representations but at much slower speeds.