Representation Theory

This chapter introduces matrix representations, i.e., homomorphic mappings of abstract groups to groups of square matrices. The behavior of the matrices under similarity transformations will lead to the concept of reducible and irreducible representations. Especially irreducible representations play an important role, e.g., in electronic structure theory each eigenstate of the Hamiltonian can be associated to a certain irreducible representation. By introducing the character of a group element (which is the trace of the representation matrix), it is shown that there is only a finite number of inequivalent irreducible representations for any finite group. The orthogonality theorem for irreducible representations is a central theorem in representation theory. The proof of the orthogonality theorem is based on the two lemmas of Schur. From the orthogonality theorem the orthogonality of the character system of the irreducible representations follows, which has important consequences, e.g. that the number of inequivalent irreducible representations is equal to the number of classes. The chapter continues with the introduction of basis functions, direct product representations and the Wigner–Eckart theorem. Finally the theory of induced representations is outlined.