Representation theory is the study of algebraic objects, such as groups and algebras, from the point of view of symmetry and invariants in linear spaces. It is a mathematical theory with a wide range of applications in group theory, combinatorics, number theory, probability, geometry and physics.
The syllabus will vary from year to year. The following is a list of optional suggested topics which together cover many basic aspects of the representation theory of finite groups, quivers, Lie groups and Lie algebras:
1. Associative algebras, group algebras, quivers and path algebras, irreducible and indecomposable representations, Schur's lemma, semisimple algebras, Jordan-Holder and Krull-Schmidt theorems, representations of finite-dimensional algebras.
2. Representations of quivers, indecomposable representations of quivers of type A1, A2, A3, D4. The triple subspace problem, simply laced root systems, reflection functors, Gabriel's theorem.
3. Lie groups, Lie algebras and enveloping algebras. Classification of semisimple Lie algebras and their representations. Quantized enveloping algebras.
4. Representations of finite groups: Maschke's theorem, duals and tensor products of representations, characters, orthogonality of characters, character tables, Burnside's theorem, induced representations and their characters (Mackey formula), Frobenius reciprocity.
5. Representations of the symmetric group and the general linear group, Schur-Weyl duality, first fundamental theorem of invariant theory.