MSc Pure Mathematics and Mathematical Logic / Course details

Representation Theory is a broad and active area of algebra, with many links to other areas of mathematics. It involves the study of algebraic objects, such as groups, rings and algebras by considering their action on vector spaces and related objects. Representation Theory can appear a very different subject depending on the algebraic objects being studied, but there are common concepts shared by all. This unit will cover some of these fundamental concepts, as well as showing some of the distinctive traits of the representation theory of different algebraic objects. There will be a particular emphasis on the representation theory of finite groups.  

We introduce the analysis of the structure of modules for algebras, before specialising to group algebras, where we go into more detail. Amongst other things, we cover semisimple modules, the Artin-Wedderburn Theorem, the radical of a module, induced modules, and introduce character theory of finite groups. 

The unit will form an introduction to representation theory, with particular emphasis on the representations of finite groups. It will cover key concepts and results used to analyse modules for algebras in a variety of settings, and lay foundations for further study. 

On the successful completion of the course, students will be able to: 

1. Define algebras, modules, representations and related notions. Prove elementary properties and apply in standard examples.
2. State, prove and apply Maschke’s theorem.
3. Define and apply basic properties of homomorphisms of modules.
4. State, prove and apply the Artin-Wedderburn Theorem.
5. Define group algebras. Recognise and prove key properties. Compute examples in both characteristic zero and prime characteristic.
6. Analyse the structure of algebras through the use of idempotents and radicals.
7. Define concepts in the character theory of finite groups, including orthogonality relations and induction. Prove fundamental results and compute examples.
8. Recognise the different properties of the representation theory of distinct algebraic objects.

Algebras, their modules and representations. Submodules, simple modules, direct sums of modules, homomorphisms of algebras and modules. Main examples, including group algebras. [2 weeks]

Maschke’s Theorem and semisimple modules. [1 week]

More on homomorphisms of algebras and modules. Schur’s Lemma. The Artin-Wedderburn Theorem. [2 weeks]

Characters of finite groups and the orthogonality relations. The number of irreducible characters for a finite group. Induced modules and characters. Permutation modules. [3 weeks]  

Analysis of the structure of an algebra as a module over itself. The radical of a module and of an algebra. Idempotents and the decomposition into indecomposable modules. [2 weeks]

Further examples and revision [1 week] 

The content will be delivered through lectures.  

There will be 3 hours of contact time per week, consisting of two lectures and a weekly 1-hour tutorial.  

Peter Webb “A course in finite Group Representation Theory”