Interpretation of the Symmetry of the Clebsch-Gordan Coefficients Discovered by Regge

Abstract

An interpretation is proposed for the intriguing symmetry of the Clebsch-Gordan coefficients discovered in 1958 by Regge. The interpretation is based on the observation that, in the reduction of the Kronecker product of two irreducible representations of an SU(2) group, there appears in a natural way another SU(2) group, which is almost independent of the original one. The Regge symmetry is interpreted as the symmetry under the interchange of these two SU(2) groups. In more picturesque language, the Regge symmetry is the symmetry under the interchange of the ``two-ness'' of the two angular momenta being added with the ``two-ness'' of SU(2). It follows from this interpretation that a symmetry of the same nature is present in the generalized Clebsch-Gordan coefficient that appears in the reduction of the Kronecker product of n (and not two, except when n = 2) irreducible representations of SU(n).