Indistinguishability in DR n-fold point-sets & their ${\mathcal{S}}_n$ -invariant dual projective mappings: limitations imposed on Racah–Wigner algebras for Liouville spin dynamics of [A] _n X multi-invariant NMR systems

The theoretic implications of democratic recoupling (DR) over identical point sets with their related $\tilde{\bf U} \times\mathcal {P}(\mathcal {S}_n)$ group actions defining Liouvillian (super)boson projective mapping on carrier space(s) is re-examined in the context of $[A]_n X,[AX]_n(SU(2) \times \mathcal {S}_n)$ (model) spin systems. In such identical point set (DR) scenerio, graph theoretic recoupling with its direct Racah–Wigner algebra (RWA) for n ≥ 3 is disallowed [Atiyah, Sutcliffe, 2002,Proc. R. Soc., Lond., A458, 1089], in favour of dual group actions (over a carrier space) and DR which yields a set of $\tilde{v}\mathcal {S}_n$ invariant-labelled disjoint carrier subspaces [Temme, 2005, Proc. R. Soc., Lond. A461, 341] in formalisms that define the {$T^k _{\{\tilde{v}\}}(11..1)$ } ’set completeness’, based on group invariants and their cardinality, |SI|⁽ⁿ⁾ as [2006, Mol. Phys., submitted MS]. Even for tensorial properties of three-fold mono-invariant spin/isospinsystems, many particle indistinguishability (identicality) poses various problems for subsequent direct use of RWA. The value of Lévi-Civitá democratic (super) operator approach is that it generates auxilary cyclic commutation properties permitting realistic extended form RWA usage. This Lévi-Civitá -based method is restricted however to three-fold identical spin problems, similar to that of Lévy-Leblond and Lévy-Nahas [1965, J. Math. Phys., 6, 1372 ]; higher index $SU(2)\times \mathcal {S}_{n\,{\ge}\, 4}$ based problems require novel $\mathcal {S}_n$ quantum physics solutions. The purpose of this communication is to stress the need for (group) compatibility between spin symmetry of the specific problem and the algebra adopted to solve it–i.e. prior to regarding any particular (group) problem as physically non-analytic.