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 2020
- Weeks 1-4: Tuesday 12-2pm and Thursday 12-2pm in Carslaw 830.
- Weeks 5-13: Lectures will be delivered online via Zoom. Join the meetings at the Canvas page scheduled for Tuesdays and Thursdays at 12pm. Tutorial exercises will be posted on this site by the beginning of every week followed by solutions by the end of the week. You are encouraged to discuss any questions on tutorials or lectures at the Ed discussion forum which has an embedded latex functionality. During the tutorial hour 1-2pm on Thursdays I'll be on standby to answer your questions in Ed.
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.
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 non-commuting 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 1-2) Basic notions: motivation, modules over associative algebras, submodules and quotients, direct sums, irreducible and indecomposable objects, Schur's lemma. [Textbook Chapter 2]
- (Weeks 3-4) General results: Characterisations of semisimplicity (complete reducibility), density theorem, Wedderburn-Artin theorem. [Textbook Sections 3.2-3.5]
- (Weeks 5-7) Representations of finite groups: Maschke's theorem, characters, Schur's orthogonality theorem, duals and tensor products, character tables, Frobenius-Schur indicators. [Textbook Chapter 4 and Section 5.1; see also Lecture Notes uploaded in Online Resources beginning from Week 5.]
- (Weeks 8-9) Induced representations, Frobenius reciprocity. [Textbook Sections 5.8-5.10]
- (Weeks 10-13) Representations of the symmetric group. [Textbook Sections 5.12-5.13]
Of the many other books dealing with these topics, the following are at a good level:
- G. D. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, second edition, 2001.
- B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, Springer-Verlag, second edition, 2001.
- J.-P. Serre, Linear representations of finite groups, translated by L. L. Scott, Springer-Verlag, 1977.
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 8 (Week 7).
- Assignment 2 will be due before midnight on Wednesday May 20 (Week 12).
|23:59 April 8||Assignment 1||20%|
Recorded Zoom lecturesare available in Canvas, click the Recorded Lectures tab.
|Tutorials||Tuesday lectures||Thursday lectures||Assessment|
|Tutorial 1 questions
Tutorial 1 solutions
|Tutorial 2 questions
Tutorial 2 solutions
|Tutorial 3 questions
Tutorial 3 solutions
|Tutorial 4 questions
Tutorial 4 solutions
|Tutorial 5 questions
Tutorial 5 solutions
|Week 5 Lecture 1||Week 5 Lecture 2||Assignment 1 - questions|
|Tutorial 6 questions
Tutorial 6 solutions
|Week 6 Lecture 1||Week 6 Lecture 2|
|Tutorial 7 questions
Tutorial 7 solutions
|Week 7 Lecture 1||Assignment 1 (20%)
Due 23:59 April 8
|Tutorial 8 questions
Tutorial 8 solutions
|Week 8 Lecture 1||Week 8 Lecture 2|
|Tutorial 9 questions
Tutorial 9 solutions
|Week 9 Lecture 1||Week 9 Lecture 2|
|Tutorial 10 questions
Tutorial 10 solutions
|Week 10 Lecture 1||Week 10 Lecture 2|
|Tutorial 11 questions
Tutorial 11 solutions
|Week 11 Lecture 1||Week 11 Lecture 2|
|Tutorial 12 questions
Tutorial 12 solutions
|Week 12 Lecture 1||Week 12 Lecture 2|
|Tutorial 13 questions
Tutorial 13 solutions
|Week 13 Lecture 1||Week 13 Lecture 2|