Representation theory

Prerequisites:

Abstract Algebra II

Course description:

The students will learn the basics of the representation theory of associative algebras over a field, as well as, as an application, the basics of the representation theory of finite groups. Representations of groups are fundamental in many parts of mathematics, such as number theory.

Learning objectives:

- Having a solid knowledge of the main basic results (and their proofs) in representation theory.

- Being able to compute character tables of finite groups in examples.

- Constructing various representations of groups or algebras.

Detailed topics covered:

Representations of associative algebras over a field, characters, semi-simple algebras, basic results, e.g. the density theorem, the Jordan-Holder theorem, the Krull-Schmidt theorem. Representations of finite groups: Maschke's theorem, orthogonality of characters, character tables, examples. Further results: the Frobenius divisibility theorem, Burnside's theorem.

Induced representations, Frobenius formula, Frobenius reciprocity, Brauer's theorem.

Further topics: representations of the symmetric group or GL2(Fp), Schur functors.