### Representation Theory WS 2020/21

#### Schedule

 Lecture: Mondays 8:00-9:30 Room 48-562 Lecturer Jun.-Prof. Dr. Caroline Lassueur Thursdays 8:00-9:30 Room 48-562 Lecturer Jun.-Prof. Dr. Caroline Lassueur Exercises: Thursdays 11:45-13:15 In OpenOLAT Assistant: Bernhard Böhmler Office hours: on appointment

Start: the 1st lecture takes place on Thursday the 29th of October.

Important: The lecture should be followed through the live stream. The streaming link / username / password will be sent to you over our OpenOLAT News function.
The videos of the lectures are uploaded in Panopto and made available in OpenOLAT directly after the lecture. However, it may take a few hours before they are processed and published by the platform!

Exam dates: 19th of Feb., 8th/23rd of March 2021

• Nov. 29: On Thursday the 3rd of December, there is no lecture.
• Oct. 31: In November the lecture must be followed through the live stream.
• Oct. 29: Slides of the introduction
• Oct. 2020: Please register in the URM system by as early as possible, so that we can contact you by e-mail (and at the latest by Friday the 30th 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

Registration: Please register in the URM system by Friday, the 30th of October 2020, noon.

The Exercises will take place in a two-week rhythm. Each Sheet will contain 6 exercices. Three of them will be discussed as examples during the tutorial. The remaining three are left to you to be solved at home and will be corrected by the assistant and solutions will be given the week after.

Schedule:
• 29th of Oct: question session over BigBlueButton. (Recap Sheet on module theory + organisation of the lecture)
From the 5th of November on:
• Even weeks: tutorial. The assistant will answer your questions on the current Exercise Sheet and give you a few hints to help you start solving the exercises.
• Odd weeks: regular exercise class. Presentation of the solutions by the students and the assistant.

Exercise Sheets:
• Sheet 0: revision sheet on modules and algebras
• Sheet 1: simple/indecomposable/semisimple modules, the Jacobson radical
• Sheet 2: idempotents, centres, group algebras, augmentation ideal
• Sheet 3: KG-modules, Maschke's Theorem, operations on groups and modules
• Sheet 4: the Mackey formula, Clifford's theorem, projective modules
• Sheet 5: relative projectivity, Greeen correspondence
• Sheet 6: p-modular systems, lifting of idempotents
The final sheet is not to be handed-in:

#### Credit points for the execrcises

It is possible to obtain an "Uebungsschein" for this lecture. In this case, please get in touch with us at the beginning of the semester.

The following criteria must be fullfilled.
• You have handed in your solutions and obtained at least 1 point in 2 Exercises in each Exercise Sheet.
• You have presented your solution for at least 3 exercises during the Exercise Sessions.
Exercises can be handed in in groups consisting of at most 4 students.

#### References and lecture notes

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

Lecture notes:
As I take this lecture in its whole for the first time in its whole, the lecture notes will progressively be available in the course of the semester.
• Conventions
• Week 1. Appendices 1 and 2. Recap of module theory [Info]
• Week 2: Chapter 1. Foundations of Representation Theory [Info]
• Week 3: Sections 6, 7 and 8: Semisimplicity of rings and modules, the Artin-Wedderburn Theorem.
• Week 4: Sections 8, 9 and 10: Semisimple algebras and their simple modules, representations of finite groups, the group algebra and its modules.
• Week 5: Sections 11, 12, 13, 14 and 15: Maschke's theorem, simple modules over splitting fields, operations on groups and modules.
• Week 6: Sections 16 and 17: the Mackey formula.
• Week 7: Sections 18, 19 and 20: Clifford theory, radical, socle, head
• Week 8: Sections 21, 22 and 23: Projective modules for the group algebra, the Cartan matrix, symmetry of the group algebra.
• Week 9: Sections 24, 25: Representations of cyclic groups, relative projectivity
• Week 10: Sections 25, 26, 27: Relative projectivity, vertices and sources, Green correspondence
• Week 11: Sections 28, 29, 30: $$p$$-permutation modules, DVRs
• Week 12: Sections 31, 32, 33: $$p$$-modular systems, idempotent lifting, Brauer reciprocity
• Week 13: Sections 34, 35: Brauer Characters, decomposition numbers
• Week 14: Section 36, 37, 38: Blocks of rings, blocks of the group algebra, defect groups, Brauer's main theorems
• Full Text (Version August 2021)

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