### Character Theory of Finite Groups SS 2019

#### Schedule

 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: Bernhard Böhmler, M.Sc.

Public holidays / Exceptional schedule:
• 14th of May: the lecture will be given by Bernhard Böhmler;
• 11th of June: there is no lecture.

Office hours: If you have questions, please feel free to come by our offices whenever you wish.

Exam Dates:
• Combination Character Theory/Quadratic number fields (with W. Hart): 13.09.2019, 24.10.2019
• Combination Character Theory/Plane Algebraic Curves (with M. Schulze): 18.09.2019, 24.10.2019
Please, register with Frau Sternike (Room 48-511).

#### Exercises

The exercise classes begin in the 2nd week of the lecture period and then take place every second week. See [KIS] for the precise dates.

The 1st exercise sheet will be uploaded after the 1st lecture. From Sheet 2 on, the exercise sheets will be uploaded on the hand-in day of the previous one.

Exercise Sheet 1. Due date: 25.04.2019, 12:00
Exercise Sheet 2. Due date: 09.05.2019, 12:00
Exercise Sheet 3. Due date: 23.05.2019, 12:00
Exercise Sheet 4. Due date: 06.06.2019, 12:00 (Exercise 16 needs the lecture of the 4th of June)
Exercise Sheet 5. Due date: 21.06.2019, 18:00
Exercise Sheet 6. Due date: 04.07.2019, 10:00

Handing in solutions: Please hand in your solution sheets by Thursdays, noon in the dedicated letter-box next to Lecture Theater 208.

Registration: Please register for the exercises in the URM system by Thursday, the 18th of April 2019, noon (Holy Thursday).

#### Übungsscheine

You obtain an "Übungsschein" if the following criteria are fulfilled:
• you have obtained minimum 50% of the points on the Exercise sheets altogether.
• you have obtained minimum 1 point in 2 Exercises (out of 4) in each of the Exercise Sheets 2 to 6.
(1 point on Sheet 1.)
• you have actively taken part to the exercise classes: attendance to the exercise classes + presenting at least one solution on the board.

Each Exercise is worth 4 points.
Exercises can be handed in in groups of maximum two students.

#### References

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.

#### Oral Exam

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 red lines of the longer proofs.
• The Exercises mentioned in the lecture are important for the understanding of the theory.
• 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.

#### Contents of the Lecture

• Linear representations and characters
• Character tables, orthogonality relations
• Burnside's $$p^aq^b$$-theorem
• Restriction, induction, inflation, tensor products
• Clifford theory

Prerequisites: elementary group theory and linear algebra