Representation Theory

Course manual 2025/2026

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 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 SU(2).

Study materials

Literature

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

Syllabus

• Lecture notes on Representation Theory (available on the canvas site)

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 dihedral groups and of SU(2).

Teaching methods

• Hoorcollege
• Werkcollege
• Lecture
• exercise class

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

Exercise classes: The student applies the theory in the lecture notes to concrete problems by solving exercises. 

Learning activities

Attendance

Attendance requirements for the program (OER - Part B):

• Active participation is expected from every student in the course component for which the student is enrolled.
• In addition to the general requirement that the student actively participates in the education, the additional requirements per component are described in the study guide. It also specifies which parts of the component have a mandatory attendance requirement.
• If a student is unable to attend a mandatory part of the program due to personal circumstances, the student must report this in writing as soon as possible to the relevant lecturer and the study advisor.
• It is not permitted to miss mandatory parts of a component if there are no personal circumstances.
• In cases of qualitatively or quantitatively insufficient participation, the examiner may exclude the student from further participation in the component or part of it. Conditions for sufficient participation are determined in advance in the study guide and on Canvas.

Assessment

Evaluation 

Evaluation for this course consists of a final exam, midterm exam and biweekly 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 midterm exam is below 5.5, the student does not pass.

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

Inspection of assessed work

The manner of inspection will be communicated via the lecturer's website.

Assignments

Homework exercises

• Biweekly 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 be challenging.  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

Honours information

There is no honours extension for this course

Additional information

Prerequisites: Linear algebra, Algebra 1 and 2.

Contact information

Coordinator

• prof. dr. Jasper Stokman

Staff

• Jasper Stokman (teacher): j.v.stokman@uva.nl
• Shin Komatsu (teaching assistent): s.komatsu@uva.nl
• Hsin-Yi Yang (teaching assistent): h.y.yang@uva.nl