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The representation theory of finite groups, which solidifies one's knowledge of group theory, is perhaps the easiest part of the general theory of symmetry. It goes back to F. Klein who considered the possibility of representing a given abstract group by a group of linear transformations (matrices) preserving the group's structure, leading mathematicians such as G. Frobenius, I.Schur, W. Burnside and H. Maschke to follow and develop the idea further. Essentially, it is a formal calculus designed to give an explicit answer to the question "What are the different ways (homomorphisms) a finite group G can occur as a group of invertible matrices over a particular field F?". The link between group representations over a field F and modules is obtained using the concept of a group ring F[G], thus an essential step is the systematic study and classification of group rings (the so-called semisimple algebras) which behave like products of matrix rings. Therefore, the story of the representation theory of a group is the theory of all F[G]-modules, viz modules over the group ring of G over F. The ultimate goal of this course is to teach students how to construct complex representations for popular groups as well as their character tables which serve as invariants for group rings. In addition to the obvious applications to physical symmetry, the theory leads to significant insights into the structure theory of finite groups.
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 8th April 2024.
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