Cayleyan n-encoded SU(2)×n↓ embeddings: Nuclear spin permutation symmetries via polyhedral lattice-point models, for modulo-i χ(i(n↓)) combinatorial invariance sets
✍ Scribed by F. P. Temme
- Publisher
- John Wiley and Sons
- Year
- 2000
- Tongue
- English
- Weight
- 506 KB
- Volume
- 78
- Category
- Article
- ISSN
- 0020-7608
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✦ Synopsis
The complete nuclear permutational (CNP) statistics of SU(2) × S 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 (S n ↓ G)] 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 = /G/. Such realizations correspond to S 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) × S n ↓ G embedding. Both the question of the sufficiency of Cayley criterion for dual group embeddings beyond SU(m ≥ 3) × S 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 S 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 [