# MATH4314 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

Starting from Week 2:

• Lectures on Tuesday 12-2pm will be delivered online via Zoom. Join the meetings at the Canvas Zoom page.
• Lecture on Thursday 12-1pm will be in ABS Seminar Room 3170.
• Tutorial on Thursday 1-2pm will be in ABS Seminar Room 3170.

## 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 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]
• (Week 8) Induced representations, Frobenius reciprocity. [Textbook Sections 5.8-5.10]
• (Weeks 9-13) Representations of the symmetric group. [Textbook Sections 5.12-5.17, 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, Springer-Verlag 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, Springer-Verlag, second edition, 2001.
• J.-P. Serre, Linear representations of finite groups, translated by L. L. Scott, Springer-Verlag, 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).

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*DescriptionBetter markWeighting
23:59 April 21 Assignment 1 20%
23:59 May 26 Assignment 2 20%
13:30 June 17 Final exam 60%
All dates are given in Sydney time.

## Exam

will be online on 17 June, the questions will be released at 11:00am and the answers will be due at 1:30pm.

## Online resources

Tuesday lecture Thursday lecture Tutorials Assessment Lecture 1-1 Lecture 1-2 Tutorial 1 questionsTutorial 1 solutions Lecture 2-1 Lecture 2-2 Tutorial 2 questionsTutorial 2 solutions Lecture 3-1 Lecture 3-2 Tutorial 3 questionsTutorial 3 solutions Lecture 4-1 Lecture 4-2 Tutorial 4 questionsTutorial 4 solutions Lecture 5-1 Lecture 5-2 Tutorial 5 questionsTutorial 5 solutions Mid-semester break Lecture 6-1 Lecture 6-2 Tutorial 6 questionsTutorial 6 solutions Assignment 1 - questions Lecture 7-1 Lecture 7-2 Tutorial 7 questionsTutorial 7 solutions Assignment 1 (20%) Due 23:59 April 21 Lecture 8-1 Lecture 8-2 Tutorial 8 questionsTutorial 8 solutions Assignment 1 - solutions Lecture 9-1 Lecture 9-2 Tutorial 9 questionsTutorial 9 solutions Assignment 2 - questions Lecture 10-1 Lecture 10-2 Tutorial 10 questionsTutorial 10 solutions Lecture 11-1 Lecture 11-2 Tutorial 11 questionsTutorial 11 solutions Lecture 12-1 Lecture 12-2 Tutorial 12 questionsTutorial 12 solutions Assignment 2 (20%) Due 23:59 May 26Assignment 2 - solutions Lecture 13-1 Lecture 13-2 Tutorial 13 questionsTutorial 13 solutions