## Admission requirements

Linear Algebra 1 and 2, Algebra 1.

Recommended: Algebra 2.

## Description

Representation theory is about understanding and exploiting symmetry 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. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles.

After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group. One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers.

## Course Objectives

- Recall basics of ring theory.
- Basics in the theory of modules.
- Basics in homological algebra, Wedderburn--Artin and Maschke's theorem.
- The basic theory of group representations, characters and character tables.
- Burnside's theorem.

## Timetable

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## Mode of instruction

Lectures, tutorials and assessed homework.

## Assessment method

The final grade consists of homework (20%) and a written (retake) exam (80%). To pass the course, the grade for the (retake) exam should be at least 5 and the (unrounded) weighted average of the two partial grades at least 5.5. No minimum grade is required for the homework in order to take the exam or to pass the course. The homework counts as a practical and there is no retake for it; it consists of weekly assignments, of which the lowest two grades are dropped.

## Reading list

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.)

Joseph Rotmanm, Advanced Modern Algebra, Part 2. American Mathematical Society, 2017.

## Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

See the Brightspace page