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Modelling of the properties of high-spin isotopomers, as polyhedra- on-lattice-points which yield various symbolic-computational \({\mathcal{S}_{12} }\)-encodings of nuclear permutation (upto some specific SU(m) branching level), is important in deriving the spin-ensemble weightings of clusters, or cage-molecules. The mathematical determinacies of these, obtained here for higher m-valued \(SU\left( m \right) \times \mathcal{S}_{12} \downarrow \mathcal{I}\) group embeddings, are compared with that of an established group embedding, in order to collate the spin physics of [11BH]122−\(\left( {SU\left( {2\left( {m \leqslant 4} \right)} \right) \times \mathcal{S}_{12} \downarrow \mathcal{I}} \right)\) with that for [10BH]122− (SU(m ≤ 7) × ..)-analogue. The most symmetrical form of \(\left[ {\left( {^{10} BH} \right) \left( {^{11} BH} \right)} \right]_6^{2 - } \left( {\left( {\mathcal{S}_6 \otimes \mathcal{S}_6 } \right) \downarrow \left( {\mathcal{S}_3 \otimes \mathcal{S}_3 } \right)} \right)\) anion provides a pertinent example of the \(SU\left( {m > n} \right) \times \mathcal{S}_n \downarrow \mathcal{G}\) physics discussed in [10]. Retention of determinacy in the two \(\mathcal{S}_{12} \downarrow \mathcal{I}\) cases is correlated to the completeness of the 1:1 bijective maps for natural embeddings of automorphic dual group NMR spin symmetries. The Kostka transformational coefficients of a suitable model (\(\mathcal{S}_n\) module, Schur fn.) play a important role. Our findings demonstrate that determinacy persists (to \(SU\left( {m \sim {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\mathcal{S}_n\) branching levels) more readily for embeddings derived from (automorphic) finite groups dominated by odd-permutational class algebras, such as the above \(\mathcal{S}_{12} \downarrow \mathcal{I}\), or the \(SU\left( {m \leqslant 3} \right) \times \mathcal{S}_6 \downarrow \mathcal{D}_3\) case discussed in [16a,15,3d], compared to other examples — (e.g. as respectively, in press, and in [17b]): \(SU\left( m \right) \times \mathcal{S}_8 \downarrow \mathcal{D}_4\), \(SU\left( m \right) \times \mathcal{S}_{10} \downarrow \mathcal{D}_5\). Generality of the symbolic algorithmic difference approach is stressed throughout and the corresponding dodecahedral \(SU\left( m \right) \times \mathcal{S}_{20} \downarrow \mathcal{I}\) maps are outlined briefly — for the wider applicability of SF-difference mappings, or of comparable \(\mathcal{S}_n\)-symbolic methods, (e.g.) via [7]. |
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