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 14: Tuesday 122pm and Thursday 122pm in Carslaw 830.
 Weeks 513: 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 12pm on Thursdays I'll be on standby to answer your questions in Ed.
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; see also Lecture Notes uploaded in Online Resources beginning from Week 5.]
 (Weeks 89) Induced representations, Frobenius reciprocity. [Textbook Sections 5.85.10]
 (Weeks 1013) Representations of the symmetric group. [Textbook Sections 5.125.13]
Other References
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, 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 8 (Week 7).
 Assignment 2 will be due before midnight on Wednesday May 20 (Week 12).
Assessment
Date^{*}  Description  Better mark  Weighting 

23:59 April 8  Assignment 1  20% 
Recorded Zoom lectures
are available in Canvas, click the Recorded Lectures tab.Online resources
Tutorials  Tuesday lectures  Thursday lectures  Assessment  

Week 1 24/228/2 
Tutorial 1 questions Tutorial 1 solutions 

Week 2 2/36/3 
Tutorial 2 questions Tutorial 2 solutions 

Week 3 9/313/3 
Tutorial 3 questions Tutorial 3 solutions 

Week 4 16/320/3 
Tutorial 4 questions Tutorial 4 solutions 

Week 5 23/327/3 
Tutorial 5 questions Tutorial 5 solutions 
Week 5 Lecture 1  Week 5 Lecture 2  Assignment 1  questions 
Week 6 30/33/4 
Tutorial 6 questions Tutorial 6 solutions 
Week 6 Lecture 1  Week 6 Lecture 2  
Week 7 6/410/4 
Tutorial 7 questions Tutorial 7 solutions 
Week 7 Lecture 1  Assignment 1 (20%) Due 23:59 April 8 

Midsemester break  
Week 8 20/424/4 
Tutorial 8 questions Tutorial 8 solutions 
Week 8 Lecture 1  Week 8 Lecture 2  
Week 9 27/41/5 
Tutorial 9 questions Tutorial 9 solutions 
Week 9 Lecture 1  Week 9 Lecture 2  
Week 10 4/58/5 
Tutorial 10 questions Tutorial 10 solutions 
Week 10 Lecture 1  Week 10 Lecture 2  
Week 11 11/515/5 
Tutorial 11 questions Tutorial 11 solutions 
Week 11 Lecture 1  Week 11 Lecture 2  
Week 12 18/522/5 
Tutorial 12 questions Tutorial 12 solutions 
Week 12 Lecture 1  Week 12 Lecture 2  
Week 13 25/529/5 
Tutorial 13 questions Tutorial 13 solutions 
Week 13 Lecture 1  Week 13 Lecture 2 