Linear Algebra 1, Linear Algebra 2, Algebra 1.
Recommended: Algebra 2
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. Along the way, we will also meet categories, modules and tensor products.
1. Basics in ring theory
2. Basics in category theory
3. Basics in homological algebra, Wedderburn-Artin, Maschke's theorem
4. Group representations, characters, character tables
5. Burnside's theorem
Mode of instruction
Lectures (possibly online)
20% homework, 80% exam (except in the case of the retake exam, it will be 10% homework and 90% exam)
The homework score will be the average of the scores on the homework sets except that the two worst homework scores will be ignored.
Pavel Etingof, Introduction to Representation Theory.
American Mathematical Society, 2011.
Freely available on the author's website
See Brightspace and Website