Group theory in solid-state physics is not dead yet alias some recent developments in the use of group theory in solid-state physics

In this article a review is given of the principal applications of group theory in solid-state physics.

Some of these applications are well established, such as the simplification of the forms of tensors representing physical properties of crystals, the labelling of electronic energy band structures, and the study of the splitting of atomic or ionic energy levels in crystals. The general principles involved in these applications are discussed. However, no attempt is made to give a comprehensive review of all the work which has been done in these areas; for further details references are given to the existing literature.

The main intention of the article is to show that apart from the well-established applications, which are adequately described in the existing literature, there have been many new developments in recent years. Group theory has come to be applied to many other types of problems in solid-state physics and these applications have not been discussed extensively in the existing review and textbook literature on the subject. These applications include: the study of the symmetry, in k space, of constant energy surfaces and in particular the symmetry of the Fermi surface; the labelling and the degeneracies of dispersion relations for phonons, magnons, and other kinds of quasiparticles; selection rules for processes involving various particle or quasiparticle states in crystals; structure determination and phase transitions; the use of two-dimensional space groups for surfaces and thin films; and the problem of the symmetry of a (non-magnetic) crystal situated in a uniform external magnetic field. The treatment given in the article is not restricted to the use of the classical point groups and space groups but, where magnetic ordering is important, the appropriate generalized symmetry groups are considered.