Representation theory of finite groups


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. The aim of this course dealing with finite groups and complex vector spaces is to introduce this theory.

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. Another goal is to construct all irreducible representations of the symmetric group.

Prerequisites


Literature


Organisation


Schedule

Week 1: Definition of a representation; recap of group theory and linear algebra. notes, exercises
Week 2: Start with Chapter 3. notes, exercises (no lecture on April 15th)
Week 3: Finish with Chapter 3. notes, exercises
Week 4: Sections 4.2 and 4.3. (lecture via Zoom)
Week 5: Finish with Chapter 4. notes, exercises
Week 6: These lecture notes written by Jop Briët and Dion Gijswijt. More notes, exercises
Week 7: Start with Chapter 6. exercises
Week 8: Finish with Chapter 6. (no lecture on May 26th)
Week 9: (lectures cancelled)
Week 10: Sections 7.1 and 7.2. exercises
Week 11: Start with Chapter 8. exercises
Week 12: Finish Chapter 8. slides (lecture via Zoom on Thursday, lecture on Friday cancelled)
Week 13: Start with Chapter 10. (no lecture on June 30th, replacement lecturer on July 1st)
Week 14: Continue with Chapter 10. notes, exercises
Week 15: Finish with Chapter 10. notes

Übungsschein: Write me an email if you regularly attended the course and need one.

Exam: You can download the exam here.