Symmetry is probably the most basic and important concept is physics: consider, for example, momentum conservation which is a consequence of the translational symmetry of space. Eigenstate properties and the degeneracy of eigenvalues are governed by symmetry considerations. The concept of representation, which connects the symmetry aspects of a system to matrices and basis functions, is most useful for an understanding of the physical properties of a system, be it atomic, molecular, or a solid state one. Often, this removes the need for numerical calculations or at least greatly simplify them.

This lecture course deals with symmetry ele­ments and point groups, intro­duces group repre­sentations and discusses the most important properties of irreducible repre­sentations and their characters. Group theory is of particular importance in the quantum-mechanical treatment of molecular orbitals. Starting from a basic assignment of the irreducible repre­sentations of atomic orbi­tals, we will discuss symmetry-induced lowering of electronic degeneracies. The classification of mole­cular vibrations is used as a simple example for the application of group represen­tations. Other appli­cations are mole­cular orbitals as well as phonon and electron bands in solids. Since this is a lecture course for experimenta­lists, there will be few mathematical proofs; emphasis is put on the use of character tables and correlation tables, using many examples. Having attended the lecture course you should be able to solve, without recourse to calculations, problems such as finding out whether a particular electronic band in a solid will have to split by symmetry in different parts of the Brillouin zone, or why the interaction between specific atomic orbitals in a molecule is forbidden. We will also discuss spontaneous symmetry lowering such as the Jahn-Teller effect. The course also covers the symmetry of the full rotation (i.e. SU(2)) group.

There are many good textbooks for this important field. I will follow, for the most part, the excellent book by M.Tinkham, "Group Theory and Quantum Mechanics", McGraw-Hill 1964, and the classic book by E. Wigner, "Gruppen­theorie...", Vieweg 1931, (Vieweg Reprint  1977). A recent book with many applications is “Group theory – Application to the Physics of Condensed Matter” by Dresselhaus, Dresselhaus and Jorio, Springer-Verlag 2008.