Group Representation Theory, Spring 2016 (M3/4/5P12)

Lecture notes so far are here (updated 21/3).

The lectures are being videoed, and recordings can be found here.

Revision class

There will be a revision class on Tuesday April 26th, 12-1 in Huxley room 139.

Problem sheets.

Sheet 1 (updated 18/1) and some solutions (updated 18/4 --- expanded answer to Question 5).

Sheet 2 (updated 9/2, fixed a typo in Exercise 2) and some solutions.

Sheet 3 (updated 14/2) and some solutions.

Sheet 4 (updated 25/2) and some solutions. Some longer additional exercises (working out the character tables of A5 and S5) are here, and some solutions.

Sheet 5 (updated 18/4 --- fixed a small typo in Question 4) and some solutions (updated 24/3 --- I reworded the answer to Question 4 a little).

Tests.

There were two progress tests, on 16/02 and 15/03. Test 1 and some solutions.

Test 2 and some solutions.

There was also an alternative Test 1: Test and some solutions.

Mastery/Comprehension (M4,M5).

This is not for 3rd year undergraduates; this is the extra question that MSc and MSci students take. The topic is the theory of induced representations. Here is a handout covering the relevant material: handout (updated 9/03, fixed some typos 21/03). A problem sheet (fixed a typo in Exercise 2), and some solutions.

Class rep.

The class rep is Raphael, email address raphael.lenain13 at the usual.

What's in the course?

We'll cover the basics of representations of finite groups on complex vector spaces. The idea behind representation theory is to study algebraic objects, like groups, by studying how they can act on vector spaces. Studying the different possible actions (or representations) allows us to obtain new insights about groups themselves (e.g. Burnside's Theorem), as well as being interesting in its own right.

The course webpages for 2014 and 2015 contain lecture notes and problem sheets. The course content this year will be very similar, but I might reorganise the material a bit.

Getting in touch

Drop me (j.newton at imperial.ac.uk) a line if you have any questions.

Recommended books.

G. James and M. Liebeck, Representations and Characters of Groups. The first 19 sections of this book cover almost all of the course, apart from the final part on semisimple algebras. You should be able to access an ebook version through the Imperial library website.
J. P. Serre, Linear Representations of Finite Groups. Part I of this book gives a concise and elegant exposition of character theory.
J. L. Alperin, Local Representation Theory. The first couple of sections cover the part of the course on semisimple algebras. The rest of the book is about group representation theory in positive characteristic, which we don't cover.