Character tables of n-dimensional hyperoctahedral groups and their applications*

We have shown that powerful matrix-type-based multinomial generators together with combinatorial techniques can be applied to derive the conjugacy classes, irreducible representations and the character tables of n-dimensional hyperoctahedral or hypercube groups of order n! × 2ⁿ, which are generalisations of non-rigid molecular theory formulated by Longuet-Higgins. The hyperoctahedral groups appear in the symmetry groups of non-rigid molecules, non-rigid water clusters, and other weakly bound van der Waals complexes that exhibit rapid tunnelling motion among various low-lying surmountable potential energy minima. These groups also have applications in quantum chromodynamics in the classification of states of quarks. The associated combinatorics has applications in enumerations, nuclear spin statistics and NMR spectroscopy. Finally, the groups enhance understanding of chirality in biosystems, that is, the functionality of intrinsically disordered proteins. We have derived the character tables of hyperoctahedral groups of sixth dimension, and techniques have been outlined to generalise for higher dimensions in the form of matrix generators.