Clebsch–Gordan coefficients, viewed from different sides

A generalized theory of angular momenta has been developed over the past few years. The new results account for a substantial change in the role played by Clebsch–Gordan coefficients both in physical and in mathematical problems. This review considers two aspects of the theory of Clebsch–Gordan coefficients, which forms a part of applied group theory. First, the close relation of these coefficients with combinatorics, finite differences, special functions, complex angular momenta, projective and multidimensional geometry, topology and several other branches of mathematics are investigated. In these branches the Clebsch–Gordan coefficients manifest themselves as some new universal calculus, exceeding substantially the original framework of angular momentum theory. Second, new possibilities of applications of the Clebsch–Gordan coefficients in physics are considered. Relations between physical symmetries are studied by means of the generalized angular momentum theory which is an adequate formalism for the investigation of complicated physical systems (atoms, nuclei, molecules, hadrons, radiation); thus, e.g., it is shown how this theory can be applied to elementary particle symmetries. A brief summary of results on Clebsch–Gordan coefficients for compact groups is given in the Appendix.