Beyond simple point sets to the indistinguishable point-set tensorial limits of R–W algebras: multiple G-invariants and the carrier-map-based quantal-basis completeness of uniform NMR spin systems

Augmented quasiparticle (QP) mappings, as applied to indistinguishable point sets of (Liouvillian) democratic-recoupled (DR) tensors, provide for a 1:1 invariant labelling of the underlying (disjoint) dual projective map carrier subspaces, where the Liouville pattern basis set is defined via superboson unit-tensor actions on a null space, |∅〉〉. The co-operative-action Liouville algebras described here imply parallel limitations to Jucys graph recoupling and its related Racah–Wigner (R–W) algebras once DR indistinguishable point tensorial sets are involved, as in non-SR Sn, n 4 dominant (NMR) spin symmetry. The importance of Sn G-invariants, as labels for disjoint carrier subspaces in such automorphic spin symmetries, arises from their essential role in defining the quantal-completeness of indistinguishable point sets. From the established properties of augmented-QPs as super-bosons (Temme 2002 Int. J. Quantum Chem. 89 429) (i.e., beyond the earlier Hilbert-space-based Louck and Biedenharn boson pattern views), insight into Atiyah and Sutcliffe’s (A–S) assertions (Atiyah and Sutcliffe 2002 Proc. R. Soc. A 458 1089) on the limitations of graph recoupling theory to distinct point sets is obtained. This clarifies the wider analytic intractible of automorphic DR spin systems— beyond the Lévi–Civitá cyclic-commutation (R–W) approach (Lévy-Leblond and Lévy-Nahas 1965 J. Math. Phys. 6 1372) which holds for a mono-invariant problem. For (rotating-frame) density matrix approaches to [A]n, [A]n(X) and [AX]n(SU(2)× Sn) (dual) NMR systems, the focus is necessarily on the specialized nature of indistinguishable point sets within multiple invarianttheoretic-based, dynamical spin physics. Here, the GI(s) (GI-cardinality) constitute an important part of the dual irrep set, {Dk(Ũ) ×
̃[λ](ṽ)(P)}, with combinatorics, as a central facet of invariant theory, playing a crucial role in the concept of ‘quantal completeness’ and the impact of A–S’s assertion 1751-8113/08/015210+11$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1 J. Phys. A: Math. Theor. 41 (2008) 015210 F P Temme on the existing NMR tensorial physics. Clearly, the role of Liouvillian Yamanouchi projection, now as disjoint subspatial-based transformational properties, defines such DR bases and their unit tensors. A brief outline is given of the structure of augmented general-indexed Lévi–Civitá superoperators with their multiple GI-labelled carrier subspaces. PACS numbers: 02.10.−De, 02.10−v, 82.56.−b List of Abbreviations Employed CFP (SU(2))-based coefficient of fractional parentage DR democratic recoupling G-, GI group/group invariant H a superboson carrier space L–C Lévi–Civitá form L̂ a Liouvillian QP Quasiparticle R–W Racah–Wigner (algebra) SR (in mathematics context) simply-reducible TRI time-reversal invariance YM Yamanouchi terms (as Sn labellings) (λ n), is a number partition; |∅̃〉〉 and |kqv〉〉 represent, respectively, a null and its corresponding Liouville space basis. Wybourne suppressed leading-part notation employed in 〈r2r3〉Sn irrep and χ 〈.〉 character notation. Z(C) and Z(R) represent complex and real algebras, whereas Zn0, etc notations refer to graph schemata; ̃ is a (Gel’fand) (L) shift.