Specialised NMR spin-cluster symmetry chains under SU(m) X Ln group dualities: Role of democratic recoupling and projective geometric mapping from SU(m) p-tuples onto Γ[γ](v: Ln), or onto Γ(v: Ln↓G), for 6⩽n-fold cage clusters

The inherent nature of the dual group spin algebras for NMR spin clusters over their associated carrier spaces is examined in the context of scalar invariants (SIs). The [A]n(SU(m)×Ln↓G) aspects of [AX]n cage spin biclusters furnish these SI forms, and focus our attention on their part in the Ln-induced representational hierarchy for democratic recoupling in NMR spin-dynamical formalisms. The purpose of the present work is to examine the nature of the specialised symmetry chains, their mappings and SU2-distinct {H̃v} carrier subspaces, especially for non-“magnetically equivalent” (non-ME) supermolecular spin biclusters in the 6 ⩽ n ⩽ 20, 60, 2160 limit. A hierarchy of intracluster {Jij} interactions ensures that the Ln-group and its graph-theoretical schemata dominate the spin algebra. These Heisenberg aspects arise from the bilinear terms of the zeroth Hamiltonian, Ĥ0. Their dominance highlights certain contrasts with conventional unitary-algebras and Wigner unit-operator properties.