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 (tentative)

We will aim to do everything in Steinberg's book excluding Section 7.3, Chapter 9 and Chapter 11.

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 Briƫt and Dion Gijswijt. More notes, exercises
Week 7: Start with Chapter 6. exercises
Week 8: (no lecture on May 26th)
Week 9:
Week 10:
Week 11:
Week 12: (lecture via Zoom)
Week 13: (replacement lecturer)
Week 14:
Week 15:

Exam: July 22th