Schur functions over polyhedral lattice‐point models: Contrasts in determinacy criteria for ((λ 1 λ 2 )⊢ n ) vs. ((λ 1 λ 2 ,…,λ m ≤( n /2) )⊢ n ) partite SU( m )×𝒮 n ↓𝒢 group embeddings: SU( m )×𝒮 8 ↓𝒟 4 spin symmetry of [H 11 B] \documentclass{article}\pagestyle{empty}\begin{document}$_{8}^{2-}$\end{document}

Earlier (λ 1 =λ 1 λ 2 )⊢ n bipartite modeling of n ‐fold nuclear spin (½) permutational (CNP), or NMR, systems provided an exclusive “algebraic geometric” demonstration of Cayley's criterion for mathematical determinacy of natural group embeddings. Other wider determinacy criteria are examined here, based on a comparison of (a) algorithmic n combinatorial encodings with (b) projective decompositions on “restricted (subgroup) space(s),” these being derived from m ≥3 SU( m )× n ↓ embeddings, involving (λ=(λ 1 λ 2 λ 3 ,…,λ m ))⊢ n n irreps (of, e.g., [ 2 H] n , [ 11 B] n ( n ) spin ensembles of \documentclass{article}\pagestyle{empty}\begin{document}$[\mathrm{H}\mathrm{B}]_{8}^{2-}$\end{document} ). Here, Cayley's criterion (CC) alone is no longer a sufficient requirement to ensure determinacy. Completeness of the distinct 1:1 bijective {[λ]→Γ( n ↓)} group subduction maps is shown to be a reliable general criterion [Eur‐Phys. J., B11, 177 (1999)]. It corresponds (for non‐ℐ, n subductions) to the retention of propagated (overall) self‐associacy (SA), onto the mathematical field of subgroups, a well‐know attribute of Yamanouchi (group) chains (YC) [Chem. Phys., 238, 245 (1998)]. In [Physica, A210, 435 (1994); A227, 314 (1996); J. Math. Chem., 21, 373; 24, 133 (1998)] we utilized regular polyhedral lattice‐point (PLP) models [i.e., as m ‐distinct color (labels) taken over a set of n ‐fold PLPs] to depict (λ 1 λ 2 ,…,λ m )⊢ n Schur functions (SFs) on restricted space maps [as (b) above]. On comparing such results with the algorithmic discrete mathematics of Kostka coefficients over the n ‐encoded irreps of (a) above, a fuller picture of group embedding appears. Here, {SU( m )× 8 ↓ 4 : m ≤4} embeddings are examined for a solvated eightfold 11 B‐borohydride, noting Sullivan–Siddall's [(λ 1 ,…,λ m ≡ n )⊢ n )]‐partite determinacy limit [J. Math. Phys., 33, 1642 (1992)]. Naturally, the recent role in quantum spin physics of the n group and of combinatorics (via group actions) draws on Biedenharn and Louck's (Bielefeld) views on dual groups over permutational spin space [see: Lect. Notes Chem., 12 (1979)], on symbolic computing [Kohnert et al., J. Symb. Comput., 14, 195 (1993); on SYMMETRICA library], as well as on the above SFs‐on‐PLP models for restricted space mappings. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 5–14, 2000