MA 426: Complex Representations of Finite Groups

2007--2008

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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.