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. If you are interested, please email me and I will add you to the email list.

Lectures: Tuesday and Thursday 1:10 PM -- 2:25 PM in TBA.

My office hours: 9 - 10 AM on Tuesday. If you want to come by another time during the week, please email me.

TA: Morena Porzio, office hours / help room hours: 9-10 AM and 6-8 PM in room 406 math.

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 take home midterm in the form of a longer set of exercises to take home in the midterm week (so due on Thursday, Oct 19). There will be a final exam.

Grading: Grades will be computed using scores on weekly problem sets, a midterm, and a final exam. The final exam will be worth 40% and the other 60% will be from the weekly problem sets with a higher weight for the one due on March 9 (which means it'll be worth roughly 12% and the other ones roughly 4% each).

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

1. A gentle introduction to the material and some linear algebra. To keep up, please look at the Notes on Linear Algebra (this should mostly be material you are familiar with) and the Notes on Inner Products.
2. More on hermition inner products. Definitions of representations, maps between representations, equivalence of representations, direct sums, duals, tensor products, and homs. Definition of irreducible and simple representations. Definition of completely reducible representations. Extra: Jordan forms (of invertible matrices) and classification of irreducible representations of the infinite cycle group G = Z and an example of a representation of Z which is *not* completely reducible.
3. Unitary representations. Complete reducibility of unitary representations. Representations of finite groups are unitary, hence completely reducible. Schur's lemma. Central elements act by scalars in an irreducible representation. Irreducible representations of abelian groups are always 1-dimensional.
4. Characters of representations. First properties of characters. Projector onto the invariant part. Schur orthogonality relations.
5. Examples of character tables.
6. Completeness of characters. More on character tables.
7. Character table of A_5 and some lemmas.

Problem sets:

1. Here is the first problem set due Thursday, Sep 14 in class.
2. Here is the second problem set due Thursday, Sep 21 in class.
3. Here is the third problem set due Thursday, Sep 28 in class.
4. Here is the fourth problem set due Thursday, Oct 5 in class.
5. Here is the fifth problem set due Thursday, Oct 12 in class.
6. Here is the sixth problem set due Thursday, Oct 19 in class.
7. Here is the seventh problem set due Thursday, Oct 26 in class.
8. Here is the eigth problem set due Thursday, Nov 2 in class.
9. Here is the ninth problem set due Thursday, Nov 9 in class.
10. Here is the tenth problem set due Thursday, Nov 16 in class.
11. Here is the eleventh problem set due Thursday, Nov 30 in class.
12. Here is the twelth problem set due Thursday, Dec 7 in class.

Material: Online and offline texts to use:

• Michael Harris webpage for his version of 4044.
• Mikhail Khovanov webpage for his version of 4044.
• Robert Friedman webpage for his version of 4044. Here is a list of notes from Professor Friedman's course:
  1. Notes on Linear Algebra
  2. Notes on Inner Products
  3. Notes on Group Theory
  4. Notes on Representations
  5. Notes on Characters, I
  6. Notes on Permutation Representations
  7. Notes on Characters, II
  8. Notes on the Fourier Transform
  9. Notes on Characters, III
  10. Notes on Induced Representations, I
  11. Notes on Induced Representations, II
  12. Notes on Real Representations
  13. Notes on Representations of the Symmetric Group
  14. Notes on Representations of GL_2
• Online textbooks (I'm sure there are many others):
  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.
• Example list of textbooks on the subject:
  1. J.-P. Serre, Linear representations of finite groups.
  2. B. Sagan, The symmetric group.
  3. B. Simon, Representations of finite and compact groups.
  4. G.James and M.Liebeck, Representations and characters of groups.
  The library will have more. Any book with representation theory of finite groups should be good enough. I may also have some copies myself; come and find me in my office to see.