Representation Theory of Finite Groups

Three different types of group representations have been considered together with their possible applications. These are atom and bond representations which are topologically-relevant representations of the molecular symmetry group, projective representations which, although still composed of sets of unitary matrices, obey a modified group multiplication rule and corepresentations which consist of sets of matrices half of which are unitary and half anti-unitary. Atom and bond representations are defined as reducible representations of a molecular point group which serve to describe the topological structure and composition of a molecule. They are amenable to computer storage and methods are given for resolving these representations into irredu­cible representations which correspond to equivalent sets of atoms or bonds. It is shown how bond representations can be derived from atom representations and a set of tables of both atom and bond representations is included. Appli­cation to additivity formulations of molecular properties is indicated, together with structural details of molecules and the identification of bending, stretching and redundant vibrational modes. All different representation groups of the point groups are established and their character tables presented. These enable the construction of equivalent alternative sets of projective representations as well as to provide an easy route to the determination of double and space group repre­sentations. The construction of the representation group clears up incompatabilities in already published literature on character systems for projective representations and shows that of all different methods available for the construction of these representations this one is most likely to be free from errors. The availability of alternative representation groups allows greater scope for the processes of ascent and descent in symmetry. Correlation tables are provided for the representation groups as well as tables of the symmetrized squares and cubes of projective representations. The set of single and double valued corepresentations for each black and white magnetic group is identified with the vector representation of one or two abstract groups of known structure and character table. This facilitates the construction of the character tables (complete sets of which are presented for the first time) and reveals that in those cases where one abstract group is sufficient a formal character theory for providing symmetrized powers of corepresentations can be established, contrary to recent indications. Two types of cases are found where it is convenient to transform Wigner’s corepresentation matrices and it is shown that normal group theoretical analysis can only be applied to Wigner's first type of corepresentation if his concept of physical equivalence is replaced by a group theoretical concept.