Cayleyan 𝒮 n ‐encoded SU(2)×𝒮 n ↓𝒢 embeddings: Nuclear spin permutation symmetries via polyhedral lattice‐point models, for modulo‐ i χ(𝒞 i (𝒮 n ↓𝒢)) combinatorial invariance sets

The complete nuclear permutational (CNP) statistics of SU(2)× n spin ensembles of forms [A] n /[AX] n for cage molecules [i.e., exclusive ( not mixed‐isotope) isotopomers] are shown to yleld totally analytic invariance sets on the basis of vertex‐point spins (or more generally Schur function labels) on regular polyhedral lattice‐point models and their modulo‐ i [of C i ( n ↓)] algebras in the specific cases discussed. This occurs only when (i.e., iff) they correspond to the criterion for Cayleyan group embedding or index n =//. Such realizations correspond to n symbolic (lattice‐based) encodings, well known in cybernetics. Hence exclusively combinatorial invariance descriptions [within (Voronoi) vertex‐point lattice geometric models], within j ≡0 mod( i ) (equivalence modulo‐ i ) for each of i indices of the class operators (of the embedded group) C i s, else a null factor, arise for certain specific automorphic CNP/NMR spin symmetries; to date these are shown to be limited to some five main Cayleyan types of SU(2)× n ↓ embedding. Both the question of the sufficiency of Cayley criterion for dual group embeddings beyond SU( m ≥3)× n and that of the further role of Kostka coefficient hierarchy in the natural embedding process, are reviewed here and more extensively in related work [Eur. Phys. J., B 11, 177 (1999); Int. J. Quant. Chem., 78(1), 5–14 (2000)]. A brief comment on the value of Yamanouchi chain‐based system invariants reaffirms the central role of the n group and its encodings in subduction processes associated with spin algebras and in certain fundamental Liouvillian (bosonic) mapping processes [Physica, A198, 245 (1993)]. In highlighting these linkages between subtopics, we demonstrate the ultimate consequences of Balasubramanian's use of automorphisms, based on the { J ij } zeroth‐order structure, in NMR [J. Chem. Phys., 78, 6258 (1983)], or equally his use of cycle‐index methods in CNP statistics. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 71–82, 2000