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.

- 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.

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.

- 1. I encourage you to use the web (e.g. wikipedia) to find information, but please don't refer to web pages, except if they are preprints (=preliminary version of a research paper, to be submitted to a journal). You should only cite published books, papers published in peer reviewed journals and preprints of papers that have been or will be submitted to peer reviewed journals.
- 2. The proper publication details of most books and papers in mathematics (at
least pure mathematics) can be found on mathscinet.

For example, if I type in "Serre" for author and "Linear Representations of Finite Groups" for title, then I get

"MR0450380 Reviewed Serre, Jean-Pierre 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. x+170 pp. ISBN: 0-387-90190-6 (Reviewer: W. Feit) 20CXX"

from which I extract:

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.

Or a bit more brief:

J.-P Serre, Linear representations of finite groups, Translated from the second French edition, Springer-Verlag, New York-Heidelberg, 1977. - 3. Label your references (e.g. with numbers) and make sure that each item
in the bibliography is cited at least once. You cite a book or paper whenever
you use a result from it.

Example in LaTeX (LaTeX it at least twice):\documentclass[a4paper,11 pt]{amsart} \begin{document} By \cite[Theorem~20]{Serre}.... ....... \begin{thebibliography}{99} \bibitem{Serre} J.~P.~Serre, {\it 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. \end{thebibliography} \end{document}

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

- 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.

- 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.