# Course Content

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 works.
• Representation of a group.
• Examples of representations. Trivial representation. Regular representation.
• Equivalent representations.
• Arithmetic 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;
• 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.

# Materials

Homework due October 17 [PS] [PDF]
Homework due October 31 [PS] [PDF]
Homework due November 14 [PS] [PDF]
Homework due November 21 [PS] [PDF]
Homework due December 3 [PS] [PDF]
Homework due January 16 [PS] [PDF]

Discussion of the second homework (draft version, beware of typos) [PS] [PDF]

A sample exam paper [PS] [PDF]
Solutions to the sample paper [PS] [PDF]
Further info on the exam [PS] [PDF]

The final exam paper [PS] [PDF]
Solutions to the final paper [PS] [PDF]

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).

Check timetables to find out the examination date.

# Textbooks

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.

# Assessment

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 assignments.