On mixed vs. exclusively-bosonic$\left\{ {\left[ {\tilde \lambda } \right]:p \leqslant 2^2 } \right\}$(SU2×S _n ) irrep sets over$\left\{ \tilde {\mathbb{H}}_v \right\}$carrier subspaces, in the structure of (SU2 ×S _n ⊗(SU2×S _n )^†)-derived Liouville space of NMR spin dynamics

Abstract

The SU2 ×S _n dual group algebra underlying Liouville NMR formalisms, which applies to extended spin cage-clusters [A] _n of [AX] NMR spin systems, is briefly presented in terms of its mapping and superbosonic properties over the$\left\{ \tilde {\mathbb{H}}_v \right\}$ specialised carrier subspaces. By considering further both the origins of Liouville space as an augmented space based on inner tensor products of simple Hilbert space, and also the λ_SA-self-associate forms in terms of the contrasting structural aspects of the 8,12,16 ⩽n and 17, 20, ⩽n S _n groups, it is shown that mixed boson/fermion sets of component irreps should only exist up to n <, 16. This property is traced to the need for the symmetric group irreps to span a set of$\left\{ {\left[ {\tilde \lambda } \right]:p \leqslant 2^2 } \right\}$ S _n ) irreps corresponding to SU2 branching over the augmented SU2 × ς _n , spin space. Parallels are drawn with the more general quonic algebras over simple Hilbert space.