# 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

The class times are Tuesday 12-2pm and Thursday 12-2pm in Carslaw 830. Tutorials will be run on Thursdays 1-2pm.

## 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 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)
**Induced representations, Frobenius reciprocity.** - (Weeks 11-13)
**Representations of the symmetric group.**

## 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*, 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).

## Assessment

**40% assignments, 50% exam, 10% tutorial participation.**

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)**.

Week 1 24/2-28/2 |
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Week 2 2/3-6/3 |

Week 3 9/3-13/3 |

Week 4 16/3-20/3 |

Week 5 23/3-27/3 |

Week 6 30/3-3/4 |

Week 7 6/4-10/4 |

Mid-semester break |

Week 8 20/4-24/4 |

Week 9 27/4-1/5 |

Week 10 4/5-8/5 |

Week 11 11/5-15/5 |

Week 12 18/5-22/5 |

Week 13 25/5-29/5 |