MA 426: Complex Representations of Finite Groups
The purpose of this course is to give an introduction to Representation
Theory for the case of finite groups. A really important idea in
Mathematics is that a proper theory for anything should possess enough
symmetries. Besides the pure aesthetical reasons, this allows to study
things in a more efficient way, and I hope to give you a flavour of how it
- Representation of a group.
- Examples of representations. Trivial representation. Regular
- Equivalent representations.
- Arithmetic of representations.
- Irreducible representations. Schur's lemma.
- Characters and matrix elements.
- Orthogonality relations for matrix elements and characters.
- representations of a product of two groups;
- tensor powers of a faithful representation;
- dimensions of irreducibles divide the order of the group;
- Burnside's paqb-theorem.
- Set representations. Orbits, intertwining numbers etc.
- "Maybe" topics.
- induced representations;
- representations of symmetric groups (basics);
- Schur-Weyl duality;
- Hurwitz's theorem on composition algebras.
Homework due October 17
Homework due October 31
Homework due November 14
Homework due November 21
Homework due December 3
Homework due January 16
Discussion of the second homework (draft version, beware of typos)
A sample exam paper
Solutions to the sample paper
Further info on the exam
The final exam paper
Solutions to the final paper
Compound of 5 Cubes
in Dodecahedron at Wolfram.com (was used in the lectures to prove that rotations of the
dodecahedron form the alternating group A5).
to find out the examination date.
There are lots of books on the subject. Most of the topics will be
probably covered by Representation Theory: a First Course by Fulton and Harris,
Linear Representations of Finite Groups by J.-P.Serre, Noncommutative Rings by I.N.Herstein, and
Representation Theory of Finite Groups and Associative Algebras by
Curtis and Reiner.
This is a half-year course, but the exam will be in June. You will get home
assignments each week or, sometimes, each other week. Home assignments
contribute 25% of your final course mark. More precise, the final mark is
given by maximum of (100% final exam, 25% home assignments+75% final
exam). On some days we will have tutorial-style classes where you will
have an opportunity to ask questions on the current assignment (if
needed), and also to tell others your solutions for the previous