Columbia University W4044 section 001
Representations of finite groups

Basic information

Call number: 92198
Room/Time: MW 1:10pm--2:25pm, 520 Math
Instructor: Mikhail Khovanov
Office: 517 Math
Office Hours: Walk-in or by appointment
E-mail: khovanov@math.columbia.edu
 
TA: You Qi
Discussions: Monday 9-10am, Room 622
Office Hours: Tue 6-7pm or by appointment, Room 610
E-mail: yq2121@math.columbia.edu
 
Midterm: Wednesday, Nov 4
Final: TBA
Webpage: www.math.columbia.edu/~khovanov/finite
 

Resources

The following two freely available online textbooks cover the basics of representation theory of finite groups:
P.Webb, Representation Theory Book We need the first 5 sections (pages 1-62).
A.Baker, Representations of finite groups

Another excellent reference is P.Etingof et al., Introduction to representation theory, which also discusses category theory, Dynkin diagrams, and representations of quivers.

There is at least a dozen textbooks on representations of finite groups in the library---here's a sample:
J.-P. Serre, Linear representations of finite groups.
B. Sagan, The symmetric group.
B. Simon, Representations of finite and compact groups.
G.James and M.Liebeck, Representations and characters of groups.

Syllabus

Groups and their actions on sets.
Modules over rings and algebras, simple modules, Schur's lemma.
Actions of groups on vector spaces, representations.
Group algebras, modules, complete reducibility, Wedderburn's theorem.
Characters, orthogonality relations.
Tensor product of representations.
Restriction and induction.
Applications of representation theory.
Category theory.
The McKay correspondence, Dynkin diagrams, reflection groups.
Representations of the symmetric group.
Representations of quivers.

Homework

Homework 1      
Homework 2      
Homework 3      
Homework 4      
Homework 5      
Homework 6      
Homework 7      
Homework 8      

Additional online resources

Representations of groups

C.Teleman, Representation theory
P.Diaconis, Group representations in probability and statistics
W.Miller, Symmetry, Groups and Their Applications
Materials and links from a course on representation theory at Stanford University
J.Rabinoff, Fourier analysis on finite groups and Schur orthogonality
Wiki page on reps of finite groups
Jeremy Rickard's webpage for a rep theory course contains problem sets and solutions.
Exercises for Joseph Bernstein's course

Rings and modules

Class notes for Modern Algebra, by Matt Douglass, especially Chapters 5 and 6 of Section 1.
Notes on Algebra, by Marc Culler Treats rings, modules, modules over PIDs.
A lecture by Daniel Katz on free modules
Condensed review of groups, rings, and modules by Roy Smith
Notes on modules and rings by William Schmitt
A lecture by Mantilla Soler on (right) modules
Wikipedia's entry on modules
Notes by Ralph Howard

Miscellaneous

Zorn's lemma links: Wiki 1 2
Category theory links: 1 2 3 4 5