Representation theory of groups and algebras - Einzelansicht

The aim of representation theory is to understand the abstract structure of a group by looking at how it acts on linear spaces. In other words, we translate the problem of understanding an abstract object (the group) into a somewhat more concrete object (a group whose elements are square matrices) and exploit many tools from linear algebra to deduce some properties of the starting group. The concept of representation can be generalized to several other algebraic objects (associative algebras, Lie algebras, algebraic groups, quivers, ...).

REFERENCES. The main reference will be the lecture notes. We will mainly follow the book "A Journey Through Representation Theory - From Finite Groups to Quivers via Algebras" by C. Gruson and V. Serganova (DOI https://doi.org/10.1007/978-3-319-98271-7) Chapters 1, 2, 5, 7 (if time permits).

 LANGUAGE. The course will be held in English.

PREREQUISITES. Linear algebra and basics about groups and group actions.

CONTENTS. In this course we will cover the following topics:

- representation theory of finite groups

- character theory (we will see that tables of irreducible characters encode the essential information about representations of finite groups)

- finite-dimensional algebras

- if time permits: introduction to representations of quivers (i.e. finite oriented graphs)

The course is mainly directed at Master students in Mathematics but some students in Physics might be interested too.