Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness 1

摘要

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schönhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P3P2 wreath product group with a different assignment of positive and negative operations than the octahedron