Modular representation theory is the study of representations of groups and related algebras over fields of positive characteristic. It was initially developed by R. Brauer, with a view towards the structure of finite groups. The aim of this mini-course is two-fold: first, provide the participants with an introduction to this topic with a focus on finite groups, and second, investigate the connections between ordinary and modular representation theory.

The first part of the course we will review basic notions: representations, characters, the group algebra and its modules, modular systems, Brauer characters, vertices and sources, block algebras.

The second part will focus on current research problems in block theory. We will investigate different types of categorical equivalences between blocks of finite groups and describe their numerical and group-theoretical invariants. Finally, we will review fundamental open problems and deep conjectures moving the topic forward.

The following short notes provide a short recap on elementary concepts of module theory, which I will use throughout the lectures. It would be great if all participants could have a look at them before the start of the course.

• Modules and algebras: basics
• The chain conditions, Jordan-Hölder, composition series
• The Jacobson radical
• Simplicity, indecomposability, semisimplicity

Here you find the handouts of my three lectures. (Warning: the files are heavy!)

• MONDAY:
  • Introduction
  • Lecture 1 (Chap. 1: Representations of Finite Groups / Chap. 2: Simplicity and Semisimplicity)
• TUESDAY: Lecture 2 (Chapter 3: Indecomposable Modules)
• WEDNESDAY: Lecture 3 (Chapter 4: Brauer Characters and p-Modular Systems, Chapter 5: Equivalences of Block Algebras)

Lecture Notes will be uploaded here at the end of each lecture.

• MONDAY: Lecture 1 (Chap. 1: Representations of Finite Groups / Chap. 2: Simplicity and Semisimplicity)
• TUESDAY: Lecture 2 (Chapter 3: Indecomposable Modules)
• WEDNESDAY: Lecture 3 (Chapter 4: p-modular systems and Brauer Characters, Chapter 5: Block Theory)

FULL TEXT