Automorphic NMR spin symmetries of cage structures: exclusively combinatorial correlative mappings for specific ( n ≡ natural embedding pertinent to higher t− 13C-fullerenes

A strong correspondence is shown to exist between spin cage-cluster systems to which Cayleyʹs theorem applies, such as [13C]60z2 for z integer higher fullerenes, and the occurrence of exclusively combinatorial [χi] ( n ↓ ) invariance (ECI) sets which determine their NMR spin symmetry. The n = 60 z2, z a small integer, higher t-icosahedral cage clusters, [X13C]n and various further nested-endohedral forms represent a wide range of model systems. Their [[λ] → Γ ( n ↓ 5 ≡ )]correlative mapping aspects follow directly from the nature of the monocluster ECI sets. These allow one to derive the SU2 × n ↓ 5 spin symmetry for (bi) cluster NMR of 13C-fullerene compounds, and also the generalised wreath-product spin symmetry for endohedral “nano-onionic”-structured fullerenes. Further aspects of the “n versus sub-group cardinality” Cayley relationship for spin cage clusters, inherent in the spin symmetry automorphisms onto certain (pseudo) regular (t-)polyhedral “geometric solids”, are examined in the context of the direct natural n (≡ ) embeddings.