Representation Theory

Owner	Bachelor Wiskunde
Coordinator	Eric Opdam
Part of	Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Bachelor Wiskunde, year 3Dubbele bachelor Wis- en Natuurkunde, year 3

Course manual 2021/2022

Course content

The main topic of this course is the study of representations of finite groups. A representation of a group is a realization of the group by means of linear transformations. A good example is given by the dihedral group acting as symmetries of a regular polygon. Representations are important in many areas of mathematics, such as analysis, geometry and mathematical physics. Central questions in this field are: what are the "fundamental" representations -called "irreducible- of a group that can occur, and how can one decompose an arbitrary representation into irreducibles? 

In this course, the following topics will be covered:

• Representations of algebras and groups
• Sub-representations, irreducible representations, Schur's lemma, semisimple representations/algebras
• Tensor product of representations, 
• Maschke's Theorem,
• Characters, class functions, character table of group representations,
• Orthogonality relations of matrix coefficients and characters of group representations,
• Representations of abelian groups and product groups,
• Induced representations and Frobenius reciprocity,
• Mackey's irreducibility criterium,
• Representation theory of the symmetric group,
• Random walks on abelian groups.

Study materials

Literature

• B. Steinberg, "Representation Theory of Finite Groups. An introductory approach", Universitext. Springer, New York, 2012. ISBN: 978-1-4614-0775-1.

Other

• Four supplements to the book (for week 40, 44, 45, 47), downloadable from the blackboard page of the course. 

Objectives

• Students can work with the character table of a finite group, and construct the irreducible characters in elementary situations. 
• Students can use orthogonality relations to decompose a complex character into a sum of irreducible characters. 
• Students can use the Fourier transform in the context of arbitrary finite groups.
• Students can construct representations using induction, restriction and techniques from (multi-)linear algebra. 
• Students can classify the irreducible representations of the symmetric group using Young tableaux.
• Students can apply results of the theory of complex representations to random walks on finite groups. 

Teaching methods

• Hoorcollege
• Werkcollege
• Lecture
• exercise class

Lecture: the material is presented during the lecture to prepare the student for the theory in the book by means of self-study.

Exercise classes: The student applies the theory in the book in concrete problems by solving exercises. 

 

Learning activities

Attendance

Programme's requirements concerning attendance (OER-B):

• Each student is expected to actively participate in the course for which he/she is registered.
• If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
• It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
• In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
• In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article A-6.4, an other arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight	Details
Final grade
20% Deeltoets
70% Eindtoets
10% homework
Final grade after retake
100% Retake exam

Evaluation 

Evaluation for this course consists of a final exam, midterm exam and regular homework exercises. 

If the weighted average of the grades for the final exam and the midterm exam is above 5.5, the final grade is determined by the final exam (70%), the midterm exam (20%) and the homework (10%). (Important: the mid term and homework can have a negative effect on the final grade!) If the weighted average of the final exam and the projects is below 5.5, the student does not pass.

There is no retake for the projects and the homework. In case of a retake, the final grade is simply the grade for the retake exam.

 

Inspection of assessed work

The date, time and location of the inspection moment are in the DataNose timetable.

Assignments

Homework exercises

• Regular homework, to be worked out independently by the students; These assignments serve to help the student to keep up to date with the theory, and are otherwise not meant to chalenging.  The exercises are graded by the instructors of the exercise classes.

   

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Honours information

There is no honours extension for this course

Additional information

Prerequisites: Linear algebra, Algebra 1 and 2.

Processed course evaluations

Below you will find the adjustments in the course design in response to the course evaluations.

Contact information

Coordinator

• Eric Opdam