The SU(3) generalizatoin of Racah's SU(2j + 1) contains SU(2) group-subgroup embedding

Abstract

Racah showed how to embed the symmetry group, SU(2) or SO(3), of a physical system in the general unitary group SU(2j + 1), where the latter group is the most general group of linear transformations of determinant 1 that leaves invariant the inner product structure of an arbitrary state space H{sub j} of the physical system. This state space H{sub j} is at the same time the carrier space of an irreducible representation (irrep) (j) of the symmetry group. This embedding is achieved by classifying the vector space of mappings H{sub j} {yields} H{sub j} as irreducible tensor operators with respect to the underlying symmetry group. These irreducible tensors are the generators of the Lie algebra of SU(2j + 1). Racah's method is reviewed within the framework of unit tensor operators. The generalization of this technique to the symmetry group U(3) to obtain the embedding U(3) {contained in} U(n), where n = dim(m) is the dimension of an arbitrary irrep of U(3). As in the SU(2) case, the group U(3) is the symmetry group of a physical system, and U(dim(m)) is the most general group of linear transformations that preserves the inner product structure of an arbitrary state space H{sub (m)} of the system. This state space H{sub (m)} is at the same time the carrier space of irrep (m) of the symmetry group U(3). Preliminary results on the Lie algebraic vanishings of U(3) Racah coefficients in consequence of the embedding SU(3) {contained in} E{sub 6} {contained in} SU(27) are given. 61 refs