Representations of Finite Groups

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This semester I am teaching the undergraduate course on the representations of finite groups. Besides the beautiful standard theory we will discuss the relationship be the structure of invariant rings and complex reflection groups as an additional topic.

Lectures: Tuesday and Thursday 11:40 PM -- 12:55 PM in TBA. It is very important to attend the lectures and participate in the discussion.

My office hours: TBA. If you want to come by another time during the week or zoom with me, please email me.

TA: TBA, office hours / help room hours: TBA.

Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. For Linear Algebra, some topics are matrices, vector spaces, direct sums, tensor products, linear transformations, eigenvalues and eigenvectors, and canonical forms. For Modern Algebra, some topics are groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, and the structure of finitely generated abelian groups.

Exams: There will be a midterm and there will be a final exam. According to the projected exam schedule, the final will be on Thursday, Dec 18.

Grading: Grades will be computed using scores on weekly problem sets, a midterm, and a final exam. The final exam will be worth about 40% and the other 60% will be from the midterm and weekly problem sets.

Lectures: It is very important to be present during the lectures!

1. Introduction

Problem sets:

1. TBA

Material: Online and offline texts to use:

• Two good books:
  1. J.-P. Serre, Linear representations of finite groups. This book is available through Spring Link for Columbia students. It covers a different set of topics, but Part I is a very good and succinct introduction to most of the material in the course.
  2. G. Lehrer and D. Taylor, Unitary reflection groups. This has a wealth of information on what we will call complex reflection groups. We will only discuss a tiny part of this, namely, some of the material from Chapters 3 and 4.
• Course webpages and course notes
  1. Michael Harris webpage for his version of 4044.
  2. Mikhail Khovanov webpage for his version of 4044.
  3. Robert Friedman webpage for his version of 4044. Here are the notes from Professor Friedman's course: Notes on Linear Algebra, Notes on Inner Products, Notes on Group Theory, Notes on Representations, Notes on Characters I, Notes on Permutation Representations, Notes on Characters II, Notes on the Fourier Transform, Notes on Characters III, Notes on Induced Representations I, Notes on Induced Representations II, Notes on Real Representations, Notes on Representations of the Symmetric Group, and Notes on Representations of GL_2.
  4. Drew Armstrong's year-long course in reflection groups at the University of Miami. His course notes are available here and here.
• Online textbooks (there are many others). Unfortunately, I have not found one that uses exactly the same language as I will be using.
  1. P. Webb, Representation Theory Book
  2. A. Baker, Representations of finite groups
  3. A.N. Sengupta, Representations of algebras and finite groups: An Introduction
  4. D.M. Jackson, Notes on the representation theory of finite groups
  5. K. Christianson, Representations of Finite Groups Course Notes
• For more online resources, see the end of this older webpage of Mikhail Khovanov.
• Other textbooks on the subject:
  1. B. Sagan, The symmetric group.
  2. B. Simon, Representations of finite and compact groups.
  3. G.James and M.Liebeck, Representations and characters of groups.
• I have a number of books on this topic myself in my office. If you come by, I will happily lend you one.