Lecture:	Tuesdays	10:00 - 11:30	Room 48-438	Lecturer:	Jun.-Prof. Dr. Caroline Lassueur
Exercises:	Fridays	11:45 - 13:15	Room 48-438	Assistant:	Birte Johansson M.Sc.
Office hour:				See OpenOLAT

Office hours: See OpenOLAT for online office hours. You can also make appointments with me for individual online qusetion sessions.

Exam Dates: (with W. Hart und D. Pol)

• 15. Sept. 2020 (nachmittags)
• 23. Okt. 2020 (nachmittags)

Registration must be done with Frau Sternike: by E-Mail.

• 14th April: Introductory slides
• 14th April: link to OpenOLAT
• IMPORTANT: If you are interested in attending the lecture, please register in the URM system, so that we can easily contact you by e-mail.

\(LaTeX\)ed lecture notes will be uploaded here every Tuesday before 10am.

Lecture Notes:

• Foreword
• Conventions
• Notation index
• Weeks 1+2: Chapter 1: Linear representations of finite groups [Explanation]
  Your task for the 1st two weeks (14th+21st of April) is to read Chapter 1 of the Lecture Notes and write solutions for the exercises in Exercise Sheet 1.
  Note that Chapter 1 contains less material than I usually cover in two lectures. In order to facilitate your first steps in Character Theory this chapter also contains many remarks, which I usually only mention orally.
  All the exercises in Sheet 1 can be solved using the material up to Definition 3.1.
• Week 3: Appendices A and B: Modules and Algebras and
  §4 Modules over the Group Algebra [Explanation]
  Your task for the 3rd week (28th of April) is to read Appendices A and B and the Section 4 in Chapter 2 of the Lecture Notes.
  With these notes you can do Exercises 5, 6, and 7 on Exercise Sheet 2.
  To do Exercise 8 you will need the lecture of the 4th week.
  Note: in the oral exam there won't be any direct questions asked on the content of the Appendices.
• Week 4: §5 Schur's Lemma, §6 Representations of Finite Abelian Groups [Explanation]
  Your task for the 4th week (4th of May) is to read the 2nd part of Chapte8 2. With the statement of Schur's Lemma you can now do Exercise 8, which is a translation of Schur's Lemma in terms of matrix representations.
• Week 5: §7 Characters, §8 Orthogonality of Characters (part 1) [Explanation]
  Your task for the 5th week (12th of May) is to read the 1st part of Chapter 3. We introduce here the central notion of this lecture: characters.
  With the notes of this week you can do Exercises 9 to 11 and Exercise 12(2.).
• Week 6: §8 Orthogonality of Characters (part 2), §9 Consequences of the 1st Orthogoinality Relations, §10 The regular character [Explanation]
  Your task for the 6th week (19th of May) is to read the 2nd part of Chapter 3. We review here a series of elementary properties of characters following from the orthogonality of the irreducible characters.
  With the notes of this week you can now do Exercise 12(1.).
• Week 7: §11 The Character Table of a Finite Group, §12 The 2nd Orthogoinality Relations, Appendix C Tensor Products of Vector Spaces [Explanation]
  Your task for the 7th week (26th of May) is to read the 1st part of Chapter 4. We introduce the character table of a finite group and compute some easy examples.
  Appendix C contains a short introduction/recap on tensor products of vector spaces. Again if you have attended the Commutative Algebra, then this material is well-known to you. If not, then think that our aim is to define a multiplication operation for vector spaces. Next week, back to representations and characters, we will see that this multiplication corresponds to a true multiplication of character values in \(\mathbb{C}\).
  With the notes of this week you can do Exercise 13 and Exercise 14.
• Week 8: §13 Tensor Products of Representations and Characters, §14 Normal Subgroups and Inflation [Explanation]
  Your task for the 8th week (2nd of June) is to read the 2nd part of Chapter 4. We introduce here two methods to construct new characters from old ones. Firstly, we define a tensor products of representations induced by the tensor product of vector spaces, which we recapped in Appendix C, and we see that characters of such tensor products are just pointwise products of the corresponding characters. Secondly we consider the operation of inflation from a quotient group, which allows us to recover certain characters of a finite group from its quotients.
  With the notes of this week you can do Exercise 15 and Exercise 16.
• Week 9: §14 Normal Subgroups and Inflation (part 2), §15 Algebraic Integers and Character Values and Appendix D: Integrality and Algebraic Integers [Explanation]
  Your task for the 9th week (9th of June) is to read the 3rd part of Chapter 4, Appendix D and the 1st part of Chapter 5. To illustrate the methods developed in Chapter 5, as an example, we compute the character table of the symmetric group \(S_4\). In Appendix D, we recall some elementary results on integrality and algebraic integers. Finally we prove that character values are always algebraic integers.
  With the notes of this week you can do Exercises 17 to 20.
• Week 10: §16 Central Characters, §17 The Centre of a Character [Explanation]
  Your task for the 10th week (16th of June) is to read the 2nd part of Chapter 5. We introduce central characters and use them to prove that the degree of any irreducible character of a finite group \(G\) divides its order.
• Week 11: §17 The Centre of a Character (continued), §18 Burnside's \(p^aq\) Theorem [Explanation]
  Your task for the 11th week (23th of June) is to read the 3rd part of Chapter 5. In these sections we continue using our results on integrality of character/central character values in order to prove several results due to Burnside. The first ones are important criteria on zeros of characters and the third one is Burnside's \(p^aq\) theorem.
• Week 12: §19 Induction and Restriction [Explanation]
  Your task for the 12th week (30th of June) is to read the 1st part of Chapter 6. We introduce two new operations on characters: induction and restriction and compute the character table of the alternating group \(A_5\).
• Weeks 13+14: §20 Clifford Theory, §21 Theorem of Gallagher [Explanation]
  Your task for the 13th and 14th weeks (7th+14th of July) is to read the 2nd part of Chapter 6. We investigate induction and restriction of characters from and to normal subgroups.
  The last exercise sheet need not be handed in.
• Full text

In OpenOLAT you will also find sum-ups of each lecture. (Uploaded each Tuesday.)

The Exercises take place fortnightly.
Registration: Please register in the URM system by Friday, the 17th of April 2020, noon.

The exercises begin in the 2nd week of the lecture period and then take place every second week. From Sheet 2 on, the exercise sheets will be uploaded on the hand-in day of the previous one.

Exercise Sheet 1. Due date: 23.04.2020, 13:00 in OpenOLAT
Exercise Sheet 2. Due date: 07.05.2020, 13:00 in OpenOLAT
Exercise Sheet 3. Due date: WEDNESDAY 20.05.2020, 13:00 in OpenOLAT
Exercise Sheet 4. Due date: 04.06.2020, 13:00 in OpenOLAT
Exercise Sheet 5. Due date: 18.06.2020, 13:00 in OpenOLAT
Exercise Sheet 6. Due date: 02.07.2020, 13:00 in OpenOLAT
Exercise Sheet 7. The last exercise sheet need not be handed in.

Handing in solutions: you should hand in your solutions in OpenOLAT by the due date.

You obtain an "Übungsschein" if the following criteria are fulfilled:

• you have obtained at least 50% of the points on the Exercise Sheets (1 to 6) altogether.
• you have obtained at least 1 point in 2 Exercises in each Exercise Sheet (1 to 6)
  

Each exercise is worth 4 points.

Exercises can be handed in groups of two students, if you wish. However, as long as the lecture takes place online, we strongly recommend that everyone tries to hand in all exercises on their own, as it will allow you. to get a deeper understanding of the topic and make sure you are up to date.

Textbooks:

• [JL01] G. James and M. Liebeck, Representations and characters of groups. See [zbMATH].
• [Ser77] J.-P. Serre, Linear representations of finite groups. See [zbMATH].
  The original text is:
  [Ser98] J.-P. Serre, Représentations linéaires des groupes finis. See [zbMATH].
• [Isa06] M. Isaacs, Character theory of finite groups. See [zbMATH].
• [Web16] P. Webb, A course in finite group representation theory. See [zbMATH].
• [CCNPW85] J.H. Conway, R.T. Curtis, S.P. Norton, R. Parker, R.A. Wilson, Atlas of Finite Groups. Clarendon Press, Oxford, 1985.

As a complement to my lecture notes, I strongly recommend Peter Webb's book, whose pre-print version is available at his webpage.

In principle one should be able to explain the content of the lecture.

• Definitions, statements of the theorems/propositions/lemmata should be known.
• You should be able to explain short proofs as well as the main arguments of the longer proofs.
• The Exercises mentioned in the lecture are important for the understanding of the theory.
• There won't be any direct questions on the content of the Appendices.
• you should also be able to give concrete examples/counter-examples to illustrate the results.
• There will also be questions on concrete examples.
• Also be ready to write down formally the concepts and results you are explaining.

In particular you should be able to expalin in your own words everything that is contained in the sum-ups uploaded in OpenOLAT.