On \(SU(2)\;{ \times }\;\mathcal{S}_{n \geqslant 12} \) dual‐group tensorial sets and carrier spaces in the multiple invariant physics of multiquantum NMR. II. \(\;\{ \mathcal{S}_{12} \supset \cdot \cdot \cdot \supset [2]\mathcal{S}_2 \} \) pathways from Yamanouchi monomial reductions as [A]12 democratic Sn‐invariant labels

Abstract

The transformational and Liouville carrier space (LCS) properties of dual tensorial bases for [A...]n NMR spin systems are considered within the quantum physics of (super)boson quasiparticles, where the (contracted) auxiliary labels of \(\;SU(2)\;{ \times }\;\mathcal{S}_n \) tensors are derived from the Sn‐based scalar invariants (SIs). Beyond both the {\(s_i^2 \)} (super)bosons mappings of pattern‐algebra [F.P. Temme, Physica A 198 (1993) 245] and the (outer) k‐rank based sub‐structure of LCS (e.g., in terms of S12 irreps [F.P. Temme, J. Math. Chem. 27 (2000) 111 (this issue)]), now (cf. Jucys recoupling) we consider the dual group physics role of the \({\widetilde {\mathcal{V}}} \) auxiliary terms of the (SI‐related) democratic sets for all the \( \geqslant \)4‐fold multispin systems, as obtained via the \(\mathcal{S}_{n - 1} \supset \cdot \cdot \cdot \supset ([2])\mathcal{S}_2 \) Yamanouchi–Gel'fand chains (YGCs). The simple reducibility of LCS derives from the explicit role of such auxiliary labels in dual mapping. The full monomial YGC reduction coefficient sets, and their related sum rule, are given here for \(SU(2)\;{ \times }\;\mathcal{S}_{12} \) tensorial sets and the distinctness of individual reduction pathways is demonstrated. Recent enumerative work on SIs [F.P. Temme, J. Magn. Reson. (2000) (to appear)] (extending [P.L. Corio, J. Magn. Reson. 134 (1998) 131]) gives expressions for the numbers of independent S12(S20) SIs for icosa‐(dodeca)hedral spin ensembles. The search for additional insight into multiquantum evolution (or coherence transfer) from the use of \(\left\{ {{\mathcal{T}}_{\left\{ {\widetilde{\mathcal{V}}} \right\}}^{kq} \left( {1_1 ,...,1_n ;\left[ {\widetilde\lambda } \right]} \right)} \right\}\) dual bases motivates this work – cf. that of [M.C. Carravetta et al., J. Magn. Reson. 134 (1998) 131; B.C. Sanctuary, Molecular Phys. 55 (1985) 1017]. Studies of SIs, and of the origins of Cayley's group embedding theorem, highlight the need to retain the Sn group in quantum physics involving spin ensembles, cf. Corio's O(3) viewpoint. A recent lattice‐point model of 12‐fold cage isotopomers, now for general multipartite forms, has demonstrated that universal mathematical determinacy (a property for which YGCs are especially noted) also prevails in \(SU(m)\;{ \times }\;\mathcal{S}_{12} \; \downarrow \mathcal{I}\) natural group embedding [F.P. Temme, Eur. Phys. J. 11(1999) 177].