Year 4 Representation Theory

The Course Notes are going to appear here as the lectures are given; problem sheets and solutions will appear as the course progresses. Here is your picture - with apologies to Steven on the front right.


Remember 15% of the course mark will come from solutions to homework. The homework will consist of 6 hand-ins (each worth 10 points), and the best 5 will count toward your 15%. I will indicate on this webpage whenever there is a homework problem to hand in, and when its due date is.


Here is last year's exam and solutions and the solution to 4(a). The course was taught far in the past (but with a different syllabus and over a longer time) so you might find the past papers interesting. They are 1994 Paper 11 (M4/11/3), 1995 Paper 406, 1998 Paper 406, 2000 Paper 406; you can find them in JCMB library they are not on-line.



Lecture 1 (7/1/08): Revision on groups and linear algebra.


Here are some exercises: make sure you can do 4-6 and 9-10.

Hand in Question 4 (the D_4 and D_5 part only) and Question 2. Both are worth 5 marks. Due on 14/1/08.


Lecture 2 (10/1/08): Representations of groups.


Here are some exercises for this lecture and the next: try at least 1,2, 3, 4, 5, 7, 9, 11, 13, 14 (but it is best to do them all).

Hand in Question 3 and Question 14. The first is worth 3 marks; the second 7. Due on 21/1/08.



Lecture 3 (14/1/08): Subrepresentations and irreducibility .


Lecture 4 (17/1/08): Complete reducibility

Lecture 5 (24/1/08): Complete Reducibility, G-homomorphisms and Schur’s lemma


Here are some exercises on complete reducibility: try 3 and 4 and the stuff on unitary representations. Solutions

Lecture 6 (28/1/08): Abelian groups, isotypic components


Here are some exercises.

Hand in Question 2 and Question 7(a),(b). The first is worth 6 marks; the second 4. Due on 11/2/08.




Lecture 7 (31/1/08): Isotypic components


Lecture 8 (4/2/08): Characters


Lecture 9 (7/2/08): Orthogonality of characters


Here are some exercises to flex your mathematical muscles on. Solutions


Lecture 10 (11/2/08): Proof of orthogonality


Here are some exercises that are the start to all types of character calculations. Make sure you can do them!.

Hand in Questions 2,3 and 4. The first is worth 6 marks; the last two both 2. Due on 21/2/08.



Lecture 11 (18/2/08): Class functions


Lecture 12 (21/2/08): Character tables and lifting representations


Lecture 13 (28/2/08): Lifting representations and A_5 character table


Here are some exercises. They are all good!

Hand in Questions (3),(4),(5) (assuming (1) and (2)) and (8). . The first three are worth 5 marks; the last 5. Due on 10/3/08.




Lecture 14 (3/3/08): A_5 character table and induced representations



Lecture 15 (6/3/08): Induced representations


Here are some exercises. Solutions


Lecture 16 (13/3/08): Nilpotent groups


There is a handout for this lecture, featuring two technical lemmas.


Here are some exercises. Solutions

Hand in Question (3) from the sheet attached to Lecture 15 and Question (2) from the sheet for this lecture. The first is worth 3 marks; the second 7. Due no later than on 20/3/08 in the folder on my door.

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