### 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.

#### Updates and Information

- 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:

- [Web16] P. Webb,
*A course in finite group representation theory.*

See [UniBibliothek / MathSciNet] - [LP10] K. Lux and H. Pahlings,
*Representations of Groups, A Computational Approach.*

See [UniBibliothek / MathSciNet] - [Alp86] J. L. Alperin,
*Local representation theory.*See [UniBibliothek / MathSciNet] - [CR81] C. Curtis and I. Reiner,
*Methods of representation theory. Vol. I.*

See [UniBibliothek / MathSciNet] - [Ben98] D. J. Benson,
*Representations and cohomology I.*See [UniBibliothek / MathSciNet] - [NT89] H. Nagao and Y. Tsushima,
*Representations of finite groups.*

See [MathSciNet] - [Dor72] L. L. Dornhoff,
*Group representation theory. Part B: Modular representation theory.*See [UniBibliothek / MathSciNet] - [Nav98] G. Navarro,
*Characters and blocks of finite groups.*See [UniBibliothek / MathSciNet]

**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

#### 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.