Teaching
Modular Representation Theory - TCC
I will be teaching a course in modular representation theory via the TCC in the 2024/25 academic year. The course will largely follow Alperin's Local Representation Theory as a means to give a relatively quick introduction to the area. A basic understanding of group theory is required, along with the basic ring theory one would typically find in a first course in algebra. An understanding of ordinary representation theory or the theory of modules over a ring would be helpful but is not required. A vague summary of the course may be found here.
The lecture notes for the course so far (potentially including upcoming content for this week) may be found here. A list of notable changes since these notes were first put online is given below.
The board notes from various lectures are given below. All the material contained in these notes should be present in the typed notes, but if there's something you can't spot just let me know.
- Lecture 2: Full notes
- Lecture 5: board 1 (Teams camera) and board 2 (shared screen)
Vague changelog for online notes
- 28/11/2024: Added section 6 (cyclic defect groups) up to possible Brauer trees.
- 22/11/2024: Moved definition of principal block a little earlier. Added Example 5.15 to show that it is not necessary for \(C_G(D) \leq H\) in order for \(b^G\) to be defined.
- 21/11/2024: Finished section 5 (Blocks). Added Theorem 4.34.
- 14/11/2024: Added material up to start of blocks.
- 07/11/2024: Fixed a few notation errors. Added material up to Green correspondence for TI subgroups.
- 31/10/2024: Fixed a few typos in Example 3.39 (\(\operatorname{SL}_2(p)\) projectives): a few sneaky \(\otimes\) were pretending to be \(\oplus\).
- 30/10/2024: Added material up to vertices and sources.
- 24/10/2024: Fixed a few typos in example on indecomposable modules for cyclic groups. Added material up to injective modules.
- 21/10/2024: Added first part of group representations section.
- 21/10/2024: Added 2.30 -- 2.32: Definition and characterisation of a local algebra and \(M\) indecomposable \(\iff \End M\) local.
- 18/10/2024: Added definition of module extension. Corrected definition of heart: \(\mathcal{H}(M) \coloneqq \rad M / (\rad M \cap \soc M)\))
The course runs Fridays from 2pm to 4pm, starting 18th October 2024.
Assessment
For those that need assessment, there will be a take-home exam released on this page at 2pm on 13th December (1 week after the final lecture). Solutions should be returned to me via email by 10th January. If you need to be assessed for this course and have not yet told me, please send me an email before the end of the course.