Abstract. A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group K into sums of irreducible characters. The method is based on the use of elements of finite order (EFO) in K and is applicable to a number of problems, including decompositions of tensor products and various symmetry classes of tensors, as well as branching rules in group-subgroup reductions. The main feature is the construction of a decomposition matrix D, computed once and for all for a given range of problems and for a given K, which then reduces any particular splitting to a simple matrix multiplication. Determination of D requires selection of a suitable set S of conjugacy classes of EFO representing a finite subgroup of a maximal torus T of K and the evaluation of (Weyl group) orbit sums on S. In fact, the evaluation of D can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter. The use of the method is illustrated by some extensive examples of tensor product decompositions in E6. Modular arithmetic allows all computations to be performed exactly.

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