Teaching
Modular Representation Theory - TCC
I taught a course in modular representation theory via the TCC in the 2024/25 academic year. The course largely follows 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.
The main materials for the course are given below. Everything should be contained in the typed lecture notes.
- Lecture notes
- Lecture 2: Full handwritten notes
- Lecture 5: board 1 (Teams camera) and board 2 (shared screen)
Vague changelog for online notes
This is a list of any nontrivial changes made to the lecture notes since their creation. Some changes may still be made to the notes without being listed here, but they are typically purely cosmetic and certainly have no impact on the mathematics.
- 06/12/2024: Fixed typos and grammar errors, altered the definition of \(Y\) on page 54 to have \(i > 1\) rather than \(i > 0\) since the function \(\psi_0\) is not defined.
- 04/12/2024: Added introduction, at long last, and the Green Correspondence (Theorem 4.34).
- 04/12/2024: Added information and examples regarding the classification of indecomposable modules for blocks with cyclic defect groups.
- 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 final lecture of the course was on 3rd December 2024.
Assessment
The exam for the course is here. If there are any questions or issues regarding the exam, feel free to send me an email. Solutions should be returned to me via email by 10th January 2025.