SU (3)×𝒮 20 algebras for uniform spin‐1 ensembles on [ 2 H 12 C ] 20 , or [ 14 N ] 20 , dodecahedrane‐type lattices and analogous isotopomeric [ M 20 12 C 40 ] met‐carb subensembles: M ‐based cardinalities and completeness of 𝒮 20 spin irreps, via hierarchical {𝒞 λ⊢( n =20):( M ) } designs of polyhedral combinatorics *

The M ‐based hierarchy cardinalities of spin irreps for \documentclass{article}\pagestyle{empty}\begin{document}$[A]_{20}^{(I_{i}=1)}$\end{document} uniform nuclear magnetic resonance (NMR) /isotopomer spin ensembles are derived. Such ideas define the completeness of the number‐partition‐based (intermediate) combinatorial designs (on M) themselves inherent in specific λ⊢ n digit assembly combinatorial properties. Illustrative M ‐subspatial irrep subsets are derived via Schur decompositions from symbolic algorithms (Sagan, B. E. Symmetric Group: Representations, C‐Algorithms, Symmetric Functions; Wadsworth: Belmont, CA, 1991; SYMMETRICA package, as per Kerber, A.; Kohnert, A.; Lascoux, A. Symbolic Comput 1992, 14, 195). The results are discussed in the context of the independant cardinality of underlying system scalar invariants (SI) corresponding to the democratic auxiliary labels (Chem Phys 1998, 238, 245; J Math Chem in press), or projective recoupling, of 20‐fold dual tensorial sets. Landau‐like n ‐maps for fundamental terms plus statistical weighted subsidary maps yield the independent ∣ SI ∣s (see Europhys Lett in press). Geometric aspects of the dual group mappings imply that eventually large cage NMR ensembles must be governed by a local, rather than a global, symmetry, with the former related to established spectral deceptive NMR. This suggests a further role for dynamical networks in NMR, beyond that given by K. Balasubramanian (J Chem Phys 1983, 78, 6369) as implied by D. Watts (Small Worlds; Princeton Univ. Press: Princeton, NJ, 1999). © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002