### Modular Representation Theory WS 2019/20

#### Schedule

 Lecture: Mondays 10:00-11:30 Room 48-438 Tuesdays 11:45-13:15 Room 48-438 Exercises: Thursdays 11:45-13:15 Room 48-438 Assistant: Daniel Schäfer Office hours: Fridays 14:00-16:00

Lecturers: Jun.-Prof. Dr. Caroline Lassueur (Weeks 1-7), Dr. Niamh Farrell (Weeks 8-14)
Exam Dates: 6th of March / 14th of April 2020
Tuesday, 10th of December: ⚠ there is no lecture.

• 08.04.2020: we have merged Parts I and II of the Lecture Notes into one document.
• Webpage for weeks 8-14: here
• ⚠ Due to the noise generated by the construction works in the library, the first lectures take place in the following lecture rooms:
• Monday, 28th of Oct: 48-582
• Tuesday, 29th of Oct: 46-267
• Monday, 4th of Nov: 48-582
• Thursday, 12th of Dec: 11-222
• Please register in the URM system by Thursday, the 31st of October 2019, noon.

#### Prerequisites

• For local students: we will assume the content of the lectures GDM, AGS and Einführung in die Algebra. The lecture is built, so that you don't need to have attended Commutative Algebra and Character Theory of Finite Groups. However, both these lectures share common ideas with Representation Theory.
• For international students: you should have a good background knowledge in linear algebra and elementary group/ring/field theory.

#### Exercises

Registration: Please register in the URM system by Thursday, the 31st of October 2019, noon.

Exercise sessions:
• 31st of Oct: there is no exercise session in the 1st week.
• 7th of Nov: tutorial
• From 14th of Nov: regular exercise sessions

The exercise sheets will be uploaded here on Tuesdays after the lecture.

Exercise Sheet 1. Due date: 12.11.2019, 18:00
Exercise Sheet 2. Due date: 19.11.2019, 18:00
Exercise Sheet 3. Due date: 26.11.2019, 18:00
Exercise Sheet 4. Due date: 03.12.2019, 18:00
Exercise Sheet 5. Due date: 10.12.2019, 18:00
Exercise Sheet 6. Due date: 17.12.2019, 18:00

The regular Exercise Classes will start in the 2nd week. On Wednesday, 11th of April, there will be a short facultive tutorial reviewing examples of groups we will use throughout the lecture.

Due dates: please hand in your exercises in the dedicated letter-box next to Lecture Room 48-208. You are welcome to work in groups of several students if wished. -->

#### References and lecture notes

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

Lecture notes:
As we take this lecture for the first time, the lecture notes will progressively be available in the course of the semester.
• Full Text (Version: 08.04.2020)
• Week 1: Chapter 0: Background Material on Module Theory
• Week 2: Chapter 1: Foundations of Representation Theory
• Weeks 3/4: Chapter 2: The Structure of Semisimple Algebras
• Weeks 4/5: Chapter 3: Representation Theory of Finite Groups
• Week 5/6: Chapter 4: Operations on Groups and Modules
• Week 6: Chapter 5: The Mackey Formula and Clifford Theory
• Week 7: Chapter 6: Projective Modules for the Group Algebra
Part II of the lecture was held and written by N. Farrell.

#### 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 red lines 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 Chapter 0 and there won't be 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

• Algebras and modules
• 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
• Brauer characters
• Reduction modulo $$p$$
• Blocks and defect groups
• Brauer's 1st, 2nd and 3rd Main Theorems