Representation theory of finite groups

Abstract

Although much of the theory of finite-dimensional algebras had its origins in the theory of group representations, it seems simpler nowadays to develop the theory of algebras first and then use it to give an account of group representations. This theory has been a powerful tool in the study of groups, especially the modular theory (representations over a field of finite characteristic), which has played a key role in the classification of finite simple groups. The theory also has important appli­cations to physics: quantum mechanics describes physical systems by means of states which are represented by vectors in Hilbert space (infinite-dimensional complete unitary space). Any group which may act on the system, such as the rotation group or a permutation group of the constituent particles, acts by unitary trans­formations on this Hilbert space and any finite-dimensional subspace admitting the group leads to a representation of the group. If we know the irreducible repre­sentations of our group, this will often allow us to classify these spaces