## MA 426: Complex Representations of Finite Groups## 2007--2008## Back to my homepage## Course ContentThe 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 p
^{a}q^{b}-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.
## MaterialsHomework 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] The final exam 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 A 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 ## AssessmentThis 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. |