About the course structure:

  The topic Algebra is organised as a reading course. The first three weeks of meetings I will explain the very basics of the representation theory of finite groups. During this part of the course you should look in one of the three books from the literature list below (James & Liebeck, Alperin & Bell, or Serre) and by for a basic result for your first write-up (see CA1 below). The idea is that my lectures should provide you with the basic knowledge to read parts of (one of) these books. After each lecture I will indicate what you should be able to read.
  In the second part of the course we continue with the students that chose from their initial two topics Algebra as their final topic. These students will focus on a somewhat more advanced topic in one of the three mentioned books. The 5 to 6 hours during this second part of the course where I will be present will partly be 15-20 minute presentations by the students and partly question/discussion sessions.

Provisional details of assessment:

• CA1 is a short essay (max. 1000 words): Give a short write-up of at least one basic result (e.g. number of irreps=number of conjugacy classes) preceded by the relevant definitions (e.g. conjugacy class) and, if possible, a sketch of the proof. These definitions might partly be covered in my first three lectures.
• CA2 is a 10 minute presentation: This should be about something from the Algebra topic that you found most interesting and that can be understood by a general audience. All students and all lecturers for the course will be present and after your presentation students and lecturers will get the opportunity to ask you some questions.
• CA3 is a report (max 4000 words): This must be about the somewhat more advanced topic that you chose in the second part of the course. For example, Integrality properties of characters, or Induced representations, or The centre of the group algebra and the primitive idempotents. I will probably also ask you to answer a few exercises from one or more of the books from the literature list.

Note:
Most of this module will be driven by your own initiative and curiosity and the expectation is that the only lecturer-led meeting will be one hour per week. For the further topic, you will be expected to coordinate an extra hour meeting of all those studying the further topic and put in extra effort to private study for this module.

Rules for citation in your reports

Prerequisites:

• Vectors and Matrices (ECM1701)
• Numbers, Symmetries and Groups (ECM1706)
• Linear Algebra (ECM2712)
• Groups, Rings and Fields (ECM2711)

The literature for this topic:

• G. James and M. Liebeck, Representations and characters of groups, Second edition, Cambridge University Press, New York, 2001.
• Chapter 5,6 of J. L. Alperin and R. B. Bell, Groups and Representations, Graduate Texts in Mathematics 162, Springer-Verlag, New York, 1995.
• J.-P Serre, Linear representations of finite groups, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.

Further literature (NOT compulsary):

Chapter 7 of P. M. Cohn, Algebra, Vol 2, Second edition, John Wiley & Sons, Ltd., Chichester, 1989.

Chapter XVIII of S. Lang, Algebra, Revised third edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2002.

• The lecture on 8-10-2013.
  Note that we do not have complete uniqueness (only up to isomorphism):
  Let a group $G$ act trivially on $V=\mathbb R^2$ (i.e. $g\cdot v=v$ for all $v\in V$ and $g\in G$).
  Put $V_1=\{(a,0)\,|\,a\in \mathbb R\}$, $V_2=\{(0,a)\,|\,a\in \mathbb R\}$ and $V_3=\{(a,a)\,|\,a\in \mathbb R\}$.
  Then $V_1,V_2,V_3$ are irreducible $G$-modules (they are one-dimensional) and $\mathbb R^2=V_1\oplus V_2$, but also $\mathbb R^2=V_1\oplus V_3$ and $\mathbb R^2=V_2\oplus V_3$.
  Since $V_1,V_2,V_3$ are all isomorphic to each other (they are all isomorphic to the irreducible trivial representation of $G$), this does not contradict the uniqueness up to isomorphism.