

ClebschGordan coefficients, viewed from different sides^{a }Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation 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 ClebschGordan coefficients both in physical and in mathematical problems. This review considers two aspects of the theory of ClebschGordan 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 ClebschGordan coefficients manifest themselves as some new universal calculus, exceeding substantially the original framework of angular momentum theory. Second, new possibilities of applications of the ClebschGordan 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 ClebschGordan coefficients for compact groups is given in the Appendix.


