Natural embedding of A_5 ≡ J group in SU(m ≤ n) x T_(12) NMR spin algebras: Ⅱ. Role of λ T_n-module decompositions over {[λ′]} in determining [~(11)B]_(12)(SU(4) x T_(12)) aspects of [~(11)BH]_(12)~(2-), [~(11)B~2D]_(12)~(2-) exo-cage ions

摘要

Natural embeddings of the A_5 ≡ J finite group into the SU(m ≤ n/2) x T_(12) spin algebras, which are implicit in automorphisms associated with intra-cluster couplings for [A]_(12) NMR spin clusters, are examined in terms of Kostka coefficients which arise in T_(12)-module decompositions for λ ∣- n partitional models (e.g.) of [~(11)B]_(12) and [~2D]_(12) NMR spin clusters for the [~(11)BH]_(12)~(2-) and [~(11)B~2D]_(12)~(2-) cage molecular ions. In particular, certain important questions are considered which arise from the generality of the low-branching high-n module decompositions, or from comparisons with specific combinatorial-determined decompositions under Sagan's variant algorithmic approach to Young's rule. The value of these concepts to physical science is demonstrated in the ease with which they permit the evaluation of the {[λ] → Γ(T_(12) ↓ A_5)} correlative mapping, which for the [~(11)B]_(12) cluster would be tedious in the extreme if derived by conventional group theoretical techniques. Indeed, the absence of work on more than sixfold spin cluster problems is indicative of the nontractable nature of unitary approaches to these higher n-fold spin systems. Additional physical context to the work now reported on four-part branchings arises from the question of the determinacy of finite group embeddings in specific T_n spin symmetries, as discussed previously for Casimir and other invariants of SU(m = n) x T_n, 6 ≤ n ≤ 8, algebras. The T_n-module decompositional approach to determining the irreps of T_n ↓ G, which are pertinent to both NMR bases and ro-vibrational spectroscopy, yields direct physical insight into correlative mapping, without any recourse to either generators or other Schur-functicn techniques. Hence, this T_n-modular approach is of special value in understanding the molecular physics of spin clusters. Also, in principle, it allows one to examine the SU(m) x T_n models for signs of degeneracy, which would lead to indeterminacy.