# PMH5 Representation Theory

## General Information

This page relates to the Pure Mathematics Honours course "Representation Theory".

Lecturer for this course: Anthony Henderson.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

## Lecturer Contact Information

My office is Carslaw room 717 and I will often be available there to answer questions, except that I will usually be working from home on Mondays and Tuesdays. You can also email me with this link to ask questions or make an appointment.

## Class Times

The **lecture** times for the course are Wednesday 1-2pm and Thursday 11am-12pm in Carslaw 830.
There will also be **tutorials** on Thursday 10am-11am in Weeks 2,4,7,9,11,13 in Carslaw 830.

## 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 University of Sydney library catalogue page for this book. Since it is easily obtainable and not expensive, I recommend buying it. However, this is not strictly necessary, since most of the content of the book has been made freely available online by the authors, in a preprint version at this arXiv page. Beware that the numbering of sections and results in the published version differs from that in the preprint version; the references below are to the former. The published version contains interesting historical notes by S. Gerovitch which are not in the preprint version.

The textbook contains many exercises, some of which will be discussed in the tutorials. Depending on time, some sections of the textbook may be assigned for independent reading. Some supplementary notes on topics not covered, or discussed only briefly, in the textbook will be provided as the semester progresses.

## 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:**Jordan-Hölder theorem, characterisations of semisimplicity (complete reducibility), density theorem, Wedderburn-Artin theorem. [Textbook Chapter 3] - (Weeks 5-9)
**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] - (Weeks 10-13)
**Further topics:**representations of compact groups such as SU_{2}, representations of Lie algebras such as sl_{2}, representations of quantum groups such as U_{q}(sl_{2}). [Textbook Section 2.15 and elsewhere]

## Other References

Of the many other books dealing with these topics, the following are at a good level:

- A. Henderson,
*Representations of Lie algebras: an introduction through*gl_{n}, Cambridge University Press, see this library catalogue page. - G. D. James and M. Liebeck,
*Representations and characters of groups*, Cambridge University Press, second edition, see this library catalogue page. - J.-P. Serre,
*Linear representations of finite groups*, translated by L. L. Scott, Springer-Verlag, see this library catalogue page. - S. Weintraub,
*Representation theory of finite groups: algebra and arithmetic*, American Mathematical Society, see this library catalogue page.

## 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

**40% assignments, 60% exam.**

Tutorial questions will be posted in Canvas in advance of the tutorial classes, with solutions afterwards. These are not for assessment.

There will be two assignments, worth 20% each; they will be posted in Canvas two weeks before the due date. The questions will be similar in style to the exercises in the textbook. These assignments are to be submitted through Turnitin; see the question sheets for further instructions.

- Assignment 1 will be due before midnight on
**Wednesday April 11 (Week 5)**. - Assignment 2 will be due before midnight on
**Wednesday May 16 (Week 10)**.

The final exam, worth 60%, will be 2 hours plus 10 minutes reading time, closed-book but with a `formula sheet' provided listing some of the main definitions and formulas from the course. Last year's exam paper will be posted in Canvas and will be discussed in the tutorial in Week 13. This year's exam will have essentially the same format and formula sheet.

## Timetable

Show timetable / Hide timetable.