• Introduction: Introduction and definitions; Group theoretical notations
• Representations: Reducible, irreducibele, equivalent representations; Orthogonality
• relations for matrix elements and characters ; Reduction of representations and
• character tables; Restricted and induced representations
• Basis functions and supplements to representations: Transformation of functions and
• operators; Basis functions and eigenfunctions; Restricted representations and
• symmetry lowering; Projection operators; Direct product of groups and
• representations; Selection rules
• Twodimensional rotation and rotation reflection group: Introduction and
• twodimensional rotation group; Twodimensional rotation-reflection group
• Three dimensional rotation group and the group SU(2): Introduction - symmetry and
• conservation laws; The group SU(2) and SO(3) - SU(2) homomorphism; Reduction of
• direct product representations - addition of angular momenta; Wigner-Eckart theorem
• Applications of group theory - capita selecta: Vibrational problems; Inversion
• symmetry ; Translation symmetry of crystalline solids - space groups - band
• structure; Time reversal symmetry - Kramers' theorem; Spin and double groups -
• crystal field theory
Final competences:
1 Being able to recognise symmetry present in physical systems.
2 Being able to make use of the symmetry present in physical systems in a creative
way.
3 To master the mathematical methods of representation theory and to be able to
apply them in practical situations.
4 To understand and to be able to predict the degeneracy of eigenvalues and the
behaviour under symmetry.
5 To be able to identify the features and properties of physical systems related to
symmetry.
6 To know and to be able to evaluate the possibilities and limitations of group theory.
7 To have enough understanding with respect to group theoretical information in order
to use and evaluate literature data in a correct way.
8 To be able to apply group theoretical knowledge in other scientific domains as, e.g.,
Atomic and Molecular Physics, Solid State Physics, Subatomic Physics, Chemistry,
Spectroscopy, etc.
