The SU(3) Generalization of Racah’s SU(2j + 1) ⊃ 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 j of the physical system. This state space H 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 j → H 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) ⊂ 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 [m] of the system. This state space H [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) ⊂ E6 ⊂ SU(27) are given.