MM828: Representation theory (5 ECTS)

STADS: 13011201

Level
Master's level course

Teaching period
The course is offered in the spring semester.

Teacher responsible
Email: duck@imada.sdu.dk

Timetable
Show entire timetable
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Comment:
Ubegrænset deltagerantal. 4. kvartal.

Prerequisites:
None

Academic preconditions:
Basic notions from group, rings and linear algebra must be known.

Course introduction
The course starts with the notion of group representations, i.e.
homomorphisms into Matrix rings. This gives rise to group actions
and the concept of modules as well as the concept of a character
of a representation.

Next it is shown that a representation can be decomposed into
unique irreducible components, i.e. Maschkes theorem and Schur's
Lemma. Orthogonality relations of characters are derived thereafter.

Central topics are then induced representations and the connection with
central simple algebras, that is Wedderburn's structure theorem.

Expected learning outcome
At the end of the course, the student will be able to:

• reproduce definitions and results within the syllabus of the course 
• apply the theory to solve problems within the socpe of the syllabus of the course 
• argue in a mathematically correct and stringent way the steps in the solution of given problems 
• supply stringent and complete proofs for statements within the syllabus of the course 
• identify and descirbe mathematical structures within the scope of the syllabus of the course in specific examples

Subject overview
orthogonality relation, irreducible moduler, induceret repræsentation, character.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
Mandatory assignments. Pass/fail, internal evaluation by teacher.

Assessment and marking:
Oral exam. Internal examiner, graded after Danish 7 mark scale (5 ECTS).

The reexamination mode may differ from the ordinary exam.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 28 hours
Skills training phase: 21 hours, hereof:
 - Tutorials: 21 hours

Educational activities

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Remarks
The course is offered when needed, and can be taken as a core Master's course in mathematics, in the category Geometry & Topology. The mandatory exercises must be passed in order for the students to be allowed to go up to the oral exam.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.