### Representation Theory WS 2022/23

#### Schedule

 Lecture: Mondays 10:00-11:30 Room 48-538 Lecturer Jun.-Prof. Dr. Caroline Lassueur Thursdays 8:15-9:45 Room 48-538 Lecturer Jun.-Prof. Dr. Caroline Lassueur Exercises: Thursdays 11:45-13:15 Room 48-538 Assistant: Marie Roth Office hours: upon appointment

Start: the 1st lecture takes place on Monday the 24th of October 2022.

Exam dates: Friday, the 14th of April.

• 24th of Oct. 22: Slides of the introduction
• Oct. 2022: Please register in the URM system so that we can contact you by e-mail; at the latest by Friday the 28th of October.

#### Prerequisites

Preferably, you should have already attended the lectures Character Theory and Commutative Algebra (or similar if you come from another university). However, the necessary notions of character theory will be introduced/recalled in due course.

#### Exercises

The Exercises take place every week.
Exercise Sheets:

#### References and lecture notes

Literature:
The lecture is mainly based on the following textbooks:

Lecture notes:
The lecture will be based on my lecture notes from the winter semester 2020/21. However, there will be some changes, in particular a slightly different choice of topics/examples/... The notes relevant to the current week will be uploaded here every week.
Lecture Notes: Full Text WS 22/23

Please, let me know if you find typos (of all kinds). I will correct them. Comments and suggestions are also welcome.

#### Oral Exam

In principle one should be able to explain the content of the lecture notes.
• Definitions, statements of the theorems/propositions/lemmata should be known.
• You should be able to explain short proofs as well as the main ideas of the longer proofs.
• You should be able to give concrete examples/counter-examples to illustrate the results.
• There can also be questions on concrete examples.
• There won't be direct questions on the Appendix.
• The Exercises mentionend in the lecture notes are important for the understanding of the theory. Their statements should be known.
• Last but not least, we expect that you are able to write down formally the concepts and results you are explaining.

#### Contents of the Lecture

• Modules over finite-dimensional algebras
• Jordan-Hölder, Krull-Schmidt, Artin-Wedderburn
• The structure of semisimple algebras
• Induction/restriction/inflation
• Duality and tensor products
• Projective/injective modules
• Vertices/sources/Green correspondence
• $$p$$-permutation modules
• Brauer characters
• $$p$$-modular systems
• Blocks and defect groups
• Brauer's 1st, 2nd and 3rd Main Theorems