Schur functions over polyhedral lattice-point models: Contrasts in determinacy criteria for ((λ1λ2)⊢n) vs. ((λ1λ2,…,λm≤(n/2))⊢n) partite SU(m)×Sn↓G group embeddings: SU(m)×S8↓D4 spin symmetry of [H11B]\documentclass{article}\pagestyle{empty}\begin{document}$_{8}^{2-}$\end{document}

Earlier (λ1=λ1λ2)⊢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 𝒮n combinatorial encodings with (b) projective decompositions on “restricted (subgroup) space(s),” these being derived from m≥3 SU(m)×𝒮n↓𝒢 embeddings, involving (λ=(λ1λ2λ3,…,λm))⊢n 𝒮n irreps (of, e.g., [2H]n, [11B]n(𝒮n) 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(𝒮n↓𝒢)} group subduction maps is shown to be a reliable general criterion [Eur-Phys. J., B11, 177 (1999)]. It corresponds (for non-ℐ, 𝒜n 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 (λ1λ2,…,λm)⊢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 𝒮n-encoded irreps of (a) above, a fuller picture of group embedding appears. Here, {SU(m)×𝒮8↓𝒟4: m≤4} embeddings are examined for a solvated eightfold 11B-borohydride, noting Sullivan–Siddall's [(λ1,…,λm≡n)⊢n)]-partite determinacy limit [J. Math. Phys., 33, 1642 (1992)]. Naturally, the recent role in quantum spin physics of the 𝒮n 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

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