Symmetry-adapted liouville space

In examining nuclear spin dynamics of NMR spin clusters in density operator/generalized torque formalisms over |kqv〉〉 operator bases of Liouville space, it is necessary to consider the symmetry mappings and carrier spaces under a specialized group for such (ki = 1) nuclear spin clusters. The SU2 × Ln group provides the essential mappings and the form of H̃ carrier space, which allows one to: (a) draw comparisons with Hilbert space duality, and (b) outline the form of the Coleman-Kotani genealogical hierarchy under induced Ln-symmetry. The simple-reducibility of the tensor space of ki = 1 recoupled tensors depends on a consideration of the v forms under SU2 ≈ SO(3) of these Tkq(v) bases, where v spans all {K̃…}(k1-kn) forms. The use of Ln-combinatorial aspects, from the study of scalar invariants, and generalized word-length formalisms provide insight into the structure of explicit Ln/SU2 bases. The value of such Tkq(v:[λ~]) comes from the way in which they allow the partitioning of the density operator σ(Ln) for multispin NMR problems and their φkq(v: [λ~])[t] polarizations.