The purpose of this course is to give an introduction to
representation theory for the case of finite groups (and demonstrate
that most of those approaches work well for infinite compact groups). A
really important idea in mathematics is that a proper theory for
anything should possess enough symmetries, and that for studying
mathematical theories it makes sense to study their symmetries.
Representation theory is the main instrument for studying symmetries.
The course covers the following topics:
- Representation of a group. Examples of representations. Trivial representation. Regular representation.
- Equivalent representations. Arithmetics of representations. Irreducible representations. Schur's lemma.
- Characters and matrix elements. Orthogonality relations for matrix elements and characters.
- Applications. Representations of a product of two groups. Tensor powers of a faithful representation contain all irreducibles as
constituents. Dimensions of irreducibles divide the order of the group. Burnside's paqb-theorem.
- Set representations. Orbits, intertwining numbers etc.
- Representations and character table of A5.
On successful completion of this module, students will be able to:
- construct complex irreducible representations for various finite groups of small orders
- reproduce proofs of basic results that create theoretical background for dealing with group representations
- apply orthogonality relations for characters of finite groups to find multiplicities of irreducible constituents of a representation
- apply representation theoretic methods to simplify problems from other areas that ``admit symmetries''
- identify group theoretic questions arising in representation theoretic problems, and use results in group theory to solve problems on group representations.
Three main recommended textbooks are:
- Charles W. Curtis, Irving Reiner, Methods of representation theory (with applications to finite groups and orders)
- William Fulton, Joe Harris, Representation theory: a first course
- J.-P. Serre, Linear representations of finite groups
All three of these books cover (much) more topics than we shall address, but most things we discuss in class is in at least one of them. The most accessible reference might be the second one, the most concise and to the point — the third one. There are several copies of the third one which School of Maths bought for students taking this course to use, you can borrow them from me when you need them.
All the handouts for this course are downloadable below. In Question 3 of the first homework, the word 'complex' ended up missing in the handouts distributed in class ('complex' as in 'complex irreducible representation'). I apologise for that; since it was only discovered close to the deadline, this question is moved to the next home assignment.