# Representation Theory (BM)

Vak
2024-2025

Linear Algebra 1 and 2, Algebra 1, Algebra 2.

## Description

Representation theory is about understanding and exploiting symmetries using linear algebra.
The central objects of study are linear actions of groups on vector spaces. This gives rise to a
very structured and beautiful theory, which can be viewed as a generalisation of Fourier analysis
to a non-commutative setting. Representation theory plays a major role in mathematics and
physics. It provides a framework for understanding finite groups, special functions, and Lie groups
and algebras, and is the mathematical basis for the theory of elementary particles in physics.

## Course Objectives

After this course students will be able to:
1. Define and use basic concepts and tools from the theory of modules and from homological algebra.
2. Formulate and explain Wedderburn and Maschke’s theorems.
3. Give an overview of the basic theory of representations of finite groups.
4. Construct the character tables of many different finite groups.
5. Formulate and explain Burnside’s theorem.

## Timetable

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## Mode of instruction

Lectures, tutorials and assessed homework.

## Assessment method

Your final grade will consist of a combination of weekly homework assignments (considered as ‘practical = praktische oefening’), written final exam, and (possibly) a resit. The homework grade is the average of all weekly homework grades with the lowest two grades dropped. There is no retake for the homework.

Hendrik Lenstra: Representatietheorie, 2003. Lecture notes by Jeanine Daems and Willem Jan Palenstijn (in Dutch). There is an English translation by Gabriele Dalla Torre.

Serge Lang, Algebra. Springer--Verlag, 2002. (Chapter XVIII is about representation theory; book freely available through SpringerLink)

Jean--Pierre Serre, Linear Representations of Finite Groups. Springer New York, 1977. (Freely available through SpringerLink.)

William Fulton, Joe Harris, Representation Theory: A First Course. Springer New York, 2004. (Freely available through SpringerLink.)

## Registration

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

Yagna Dutta: y.dutta@math.leidenuniv.nl
Bart de Smit: b.de.smit@math.leidenuniv.nl

## Remarks

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.