In my post I’ve Applied for a Master of Mathematics Degree! I shared that I was most scared of the course in Modules and Representation Theory as I bombed that course in my undergrad degree (I got a 62).
The UNSW term ended in May and I got my results back and I got a 92 in the course! I am just so happy about this result as it shows that growth does happen over years.
When I took the course in my undergrad days, I could not even remember basic definitions such as torsion elements, let alone theorems such as the structure theorem of modules over principal ideal domains…
The topic of Modules and Representations was actually quite intriguing this time around although many of peers did not enjoy it – which I can completely empathise with as that was where I found myself 11 years ago. It was a very dense course, but we made it!
Here is the list of topics we covered:
- Week 1: Modules Basics: submodules, homomorphisms, quotient modules, isomorphism theorems, direct sums, free modules, torsion modules, simple modules, Schur’s Lemma, indecomposable modules
- Week 2: R-algebras, presentations, categories and functors, short exact sequences
- Week 3: Chain conditions, Noetherian and Artinian rings, composition series, finite length modules, Jordan-Hoelder’s Theorem
- Week 4: Finite generated modules over principal ideal domains, structure theorem, applications (to finite abelian groups, rational and Jordan canonical forms of matrices)
- Week 5: Semisimple rings, Wedderburn’s Theorem, Maschke’s Theorem
- Week 6: Tensor products of modules, group representations, 1-dimensional representations
- Week 7: Contragredient dual, tensor products of algebras and representations, induced representations
- Week 8: Characters, primitive central idempotents, orthogonality relations
- Week 9: Radicals of modules and rings, Fitting’s Lemma, Krull-Schmidt Theorem
I would say my favourite theorems of the course were Wedderburn’s Theorem and Maschke’s Theorem. It was just a beautiful result that tells us how many irreducible representations over C a finite group can have as well as their dimension.