Role of democratic SU2 x Sn dual-group carrier space and group structure in the superboson quantum-Liouville physics of identical spin ensembles: Maximal Sn tensor product reduction, via symbolic algebraic combinatorics

Structural aspects of superboson mappings and their dual group-based carrier spaces inherent in quantum-Liouville NMR formalisms are presented for their conceptual role in understanding the transformational properties and the spin dynamics of identical [A]n (n) spin ensembles or [AX]n systems, given in terms of {Tkq (u)} ? {|kqu > >} tensorial bases. Interest in the explicit democratic labelling of the Liouvillian carrier subspaces of [A]n spin ensembles, prompts an examination of the inner products (ITPs) that define the projective carrier space associated with superbosons. A weak-branching (WB) limit (for (bipartite) partitions l n, for n-indexed (SU2 ×)n) ITPs gives rise to: (i) maximal coefficient sets for [µ] [µʹ](n) irrep products (in Butlerʹs notation), under a sufficiently high (n 4µ, 4µʹ) indexed n group, or (ii) to identical numerical {c, lʹ } sequence-ordered sets of reduction coefficients over similar, displaced-sequence fields - on augmenting ITP to [µ] [µʹʹ], where µʹʹ > µʹ > µ, - or else (iii) to some (sequencedisplaced) subset of the latter. The origins of the decompositional WB limit as part of group structure may be traced to an algorithmic similarity between the Littlewood-Richardson and Young(III) combinatorial rules. Alternative approaches to the bipartite product decompositional mappings are possible, using both (subspatialrestricted) Schur-function techniques of Wybourne and 16n28 symbolic algorithmic enumerations based on the SYMMETRICA discrete-maths. package of Kerber et al. (J. Symbolic Comput. 14, 195 (1993)). The nature of maximal sets and scaling in ITP decompositions is established and recognition given to the role of combinatorics, n algorithms and the superboson algebras of the SU(2) × n group in (multiple) quantized spin physics.