Groups and Representations

I have been a class tutor for my supervisor, Prof. Andre Lukas’ course, Groups and Representations for four years. The course aims to introduce Lie groups and algebras to master’s level students taking the MMathPhys course at Oxford.

General information for example classes

Our classes will run on Weeks 4, 6 and 8 on Fridays at 1630-1800. They will be in the DWB seminar room - this is the small room on the second floor of the DWB building. When you come up the stairs from the entrance, simply take a left turn, go through the set of doors and the seminar room is the first door to your left.

Homework requirements and submission

There is a homework requirement for this course. This means that you must obtain at least 50% for half the questions you have submitted for each sheet. The homework deadline is noon on the Friday before the week the classes are held (so for the first one, tomorrow Friday the 31st). To submit your work, please go to the Assignments tab on Canvas and find the appropriate class (i.e. Groups and Representations Problem Class X (Lucas Leung)). You should be able to submit your work there. If you cannot access the Canvas page, please drop me an email.

I want to emphasise that this is not a stringent requirement – so as long as you have tried you should pass this requirement. In fact, try and attempt the questions. Even if you can’t finish it explain what you have done and attempted in your submissions – it is better that I see what you guys are thinking which helps me prepare the classes! This will also help you learn better.

How to understand the grades

My marking normally follows these rules – I have given you a grade A,B,C or F at the end of each question. I have not included any physical marks (numbers) – this is because I find it quite hard to assign marks in different sub-parts of a question. Instead you will get a letter and they roughly correspond to:

A - This is equivalent to a First-class. The attempt has shown excellent understanding of the course material, and can perform all book-work material to good extent. The attempt has shown extensive understanding of the material to use it in unfamiliar situations.

B – This is equivalent to a 2:1. The attempt has shown good understanding of the course material, albeit with minor confusions. The student can utilise some of the material in unfamiliar situations.

C – This is equivalent to a 2:2 or a Third-class. The attempt has shown good understanding of the course material, but with significant points of confusion. The student however is not quite capable to use the material in unfamiliar contexts.

F – The attempt has shown poor understanding of the material or there are significant errors/misconceptions in the presentation.

Apologies for the ‘ad-hoc’ way of marking – this kind of comes from the weird education system I grew to be familiar with.

How the classes will work

Unlike some courses the solutions to the questions will not be published. I will therefore aim to cover the solutions of the problems in the class. But if you have any questions or would like me to cover a particular topic, please email me before the class so I can prepare for it.

References and Supplementary Notes

Andre’s notes is probably a good place to start if you have no idea what’s going on – they are an excellent set of notes with loads of details in it. In terms of books, the mathematically-inclined of you should refer to Fulton and Harris. I will recommend some other books as the course goes on.

Over the years I have typed up some supplementary notes for the course. They will be distributed after classes are finished on Canvas. They are written in a weird format – some of them are quite chatty and informal but some of the sections are formal maths-supplementary sections. They are all non-examinable (so it is up to you how to use them). Please note that I will not share the solutions to the problems, so you should still aim to make notes during the class! That being said, please let me know if you have any questions (even on supplementary notes stuff) or ideas that you are curious about!

The current version for the supplementary notes is hosted here. Please send me an email if you spot any errors! (Current version updates: new tensor branching discussion + exponential map functoriality.)

Groups and Representations are fundamental to what we do in high energy physics. In fact, our group uses that on a daily basis. This is therefore a mathematically-rigorous course to introduce you to all these concepts that might be quite daunting to start with. It will be difficult but it will be worth-it. If you have any questions about the lectures/tutorials/physics or mathematics in general, feel free to email me or chat to me after the classes!

Extra Q&A Class

I will be running an extra Q&A class to answer questions about the course. This will be hosted online. If you are in my class, you should receive a Zoom link which I will later send out. The preliminary class time is set on 23 December at 1400 GMT.