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A numerical algorithm that computes the decomposition of anyfinite-dimen\-sio\-nal unitary reducible representation of a compact Lie groupis presented. The algorithm, which does not rely on an algebraic insight on thegroup structure, is inspired by quantum mechanical notions. After generatingtwo adapted states (these objects will be conveniently defined in {\bfDef.\,II.1}) and after appropriate algebraic manipulations, the algorithmreturns the block matrix structure of the representation in terms of itsirreducible components. It also provides an adapted orthonormal basis. Thealgorithm can be used to compute the Clebsch--Gordan coefficients of the tensorproduct of irreducible representations of a given compact Lie group. Theperformance of the algorithm is tested on various examples: the decompositionof the regular representation of two finite groups and the computation ofClebsch--Gordan coefficients of two examples of tensor products ofrepresentations of $SU(2)$.

A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups