Towards a system in space group representations

Abstract

The structure of representations of a group is tightly connected with the lattice of normal subgroups of that group. We show at first how the correlation between irreducible representations and their kernels can be used if we know the position of kernels in the lattice of normal subgroups. The rest of paper concerns the problems of deciphering the lattices of normal subgroups of space and subperiodic groups. We show that these lattices can be described easily for irreducible space groups. In case of reducible space groups we show how to get a partial description of the lattice with use of sublattices which are isomorphic images of lattices of subperiodic groups. Finally we show the ways for derivation and description of lattices of normal subgroups of subperiodic groups; in our case of layer and rod groups. Representative examples of lattices illustrate particular steps of the development of the whole scheme.