MATH4314 Representation Theory
This page contains information on the Honours Mathematics unit Representation Theory.
Lecturer for this course: Alexander Molev.
For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.
For enrolled students or other authorized people only, here is a link to the Canvas page for MATH4314.
Lecturer Contact Information
My office is Carslaw room 707. You can email me with this link to ask questions or make an appointment.
Class Times in 2021
 Lectures on Tuesday 122pm will be delivered online via Zoom. Join the meetings at the Canvas Zoom page.
 Lecture on Thursday 121pm will be in Eastern Avenue Seminar Room 120. It will also be streamlined via Zoom.
 Tutorial on Thursday 12pm will be in Eastern Avenue Seminar Room 120.
Textbook
Most of the course is close to parts of the following textbook:
P. Etingof et al, Introduction to Representation Theory, American Mathematical Society Student Mathematical Library, vol. 59, American Mathematical Society, Providence, RI, 2011.
Here is the link to the electronic version of the book on Etingof's web page.
Course Outline
Representation theory is a major area of algebra with applications throughout mathematics and physics. Viewed from one angle, it is the study of solutions to equations in noncommuting variables; from another angle, it is the study of linear algebra in the presence of symmetry; from a third angle, it is the study of the most tractable parts of category theory. Historically, the representation theory of finite groups was developed first, and the many applications and beautiful special features of that theory continue to recommend it as a starting point. However, it is important to appreciate the underlying principles which unify the representation theory of finite groups, Lie algebras, quivers and many other algebraic structures.
The rough outline (which may be modified as the semester progresses) is:
 (Weeks 12) Basic notions: motivation, modules over associative algebras, submodules and quotients, direct sums, irreducible and indecomposable objects, Schur's lemma. [Textbook Chapter 2]
 (Weeks 34) General results: Characterisations of semisimplicity (complete reducibility), density theorem, WedderburnArtin theorem. [Textbook Sections 3.23.5]
 (Weeks 57) Representations of finite groups: Maschke's theorem, characters, Schur's orthogonality theorem, duals and tensor products, character tables, FrobeniusSchur indicators. [Textbook Chapter 4 and Section 5.1]
 (Week 8) Induced representations, Frobenius reciprocity. [Textbook Sections 5.85.10]
 (Weeks 913) Representations of the symmetric group. [Textbook Sections 5.125.17, also the book by B. Sagan, Chapter 2]
Other References
Of the many other books dealing with these topics, the following are at a good level:
 W. Fulton, Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997.
 G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, v.682, SpringerVerlag Berlin Heidelberg, 1978.
 G. D. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, second edition, 2001.
 I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1995.
 B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, SpringerVerlag, second edition, 2001.
 J.P. Serre, Linear representations of finite groups, translated by L. L. Scott, SpringerVerlag, 1977.
Background Knowledge
The main prerequisites are a solid understanding of linear algebra (in particular the basic facts about matrices and eigenspaces), group theory and basic ring theory, as in MATH2961 Linear Mathematics and Vector Calculus (Advanced), MATH2968 Algebra (Advanced) and MATH3962 Rings, Fields and Galois Theory (Advanced).
Assessment tasks weightings
40% assignments, 60% exam.
There will be two assignments, worth 20% each. These assignments are to be submitted through Turnitin.
 Assignment 1 will be due before midnight on Wednesday April 21 (Week 7).
 Assignment 2 will be due before midnight on Wednesday May 26 (Week 12).
Assessment
Date^{*}  Description  Better mark  Weighting 

23:59 April 21  Assignment 1  20%  
23:59 May 26  Assignment 2  20% 
Exam
will be held in June.Online resources
Tuesday lecture  Thursday lecture  Tutorials  Assessment  

Week 1 1/35/3 
Lecture 11  Lecture 12  Tutorial 1 questions Tutorial 1 solutions 

Week 2 8/312/3 
Lecture 21  Lecture 22  Tutorial 2 questions Tutorial 2 solutions 

Week 3 15/319/3 
Lecture 31  Lecture 32  Tutorial 3 questions Tutorial 3 solutions 

Week 4 22/326/3 
Lecture 41  Lecture 42  Tutorial 4 questions Tutorial 4 solutions 

Week 5 29/32/4 
Lecture 51  Lecture 52  Tutorial 5 questions Tutorial 5 solutions 

Midsemester break  
Week 6 12/416/4 
Lecture 61  Lecture 62  Tutorial 6 questions Tutorial 6 solutions 

Week 7 19/423/4 
Lecture 71  Lecture 72  Tutorial 7 questions Tutorial 7 solutions 
Assignment 1 (20%) Due 23:59 April 21 
Week 8 26/430/4 
Lecture 81  Lecture 82  Tutorial 8 questions Tutorial 8 solutions 

Week 9 3/57/5 
Lecture 91  Lecture 92  Tutorial 9 questions Tutorial 9 solutions 

Week 10 10/514/5 
Lecture 101  Lecture 102  Tutorial 10 questions Tutorial 10 solutions 

Week 11 17/521/5 
Lecture 111  Lecture 112  Tutorial 11 questions Tutorial 11 solutions 

Week 12 24/528/5 
Lecture 121  Lecture 122  Tutorial 12 questions Tutorial 12 solutions 
Assignment 2 (20%) Due 23:59 May 26 
Week 13 31/54/6 
Lecture 131  Lecture 132  Tutorial 13 questions Tutorial 13 solutions 