Meeting Times: Wednesdays 7:45-9:45 PM in 528 Math
Instructor: Dmitry Zakharov
Office: 206A Math
Office Phone: (212) 854-5886
Office Hours: By appointment
Email: zakharov@math.columbia.edu
Prerequisites: Abstract algebra (in particular, group theory), linear algebra.

Course Description: The theory of representations of finite groups is an interesting and self-contained area of mathematics, arising in Galois theory and quantum mechanics, but also on its own. The basic theory consists of classifying the irreducible representations of a finite group via certain complex-valued functions on the group. This allows us to effectively decompose an arbitrary representation into irreducible ones. In addition to the basic theory, we will cover some of the following topics, according to the tastes and abilities of the audience:
  • Representations of the symmetric group
  • Representations of GL(2) over a finite field
  • The inverse problem - finite subgroups of GL(V)
  • Induced representations - Artin's theorem and Brauer's theorem
  • Representations over subfields: the reals and the rationals
  • Representations over fields of positive characteristic - an introduction to Brauer theory
  • Introduction to Lie theory - representations of SO(3)

    Literature: J. P. Serre, Linear Representations of Finite Groups, Springer Verlag, 1977. A classic text, also includes a section on Brauer theory. There is a copy on reserve in the library, as well as an unreserved French original. You can find a copy at AddAll.com for about $40.

    William Fulton and Joe Harris, Representation Theory: a First Course, Springer, 1999. An excellent book mainly dealing with Lie theory, but has a nice overview of finite groups as well. If you plan to take a course in representations of Lie groups, I strongly suggest you buy this one. I've put a copy on reserve in the library, and you can find one on AddAll.com for about $35.

    Grading: Every attending student is expected to give at least two talks, and you will be graded primarily on the quality of your preparation and presentation. I will also occasionally give homework.

    Attendance: Attendance is mandatory. Please inform me if you need to miss a lecture.
    September 11: I will give the introductory lecture and give an overview of the topic. We will organize the schedule of the first few talks and hopefully find a better time.
    September 20: Emily Tvetenstrand - Multininear algebra and representations of S_3.
    September 27: Mio Alter - Theory of Characters, Part I.
    October 4: Arun Chandrasekhar - Theory of Characters, Part II. Decomposition of the regular representation.
    October 11: Emily Tvetenstrand - Irreducible representations of S_4 and A_4.
    October 18: Mio Alter - Abelian groups, representations of the product group, induced representations, the group algebra.
    October 25: Arun Chandrasekhar - Artin's theorem and Brauer's theorem.
    November 1: Emily Tvetenstrand - Representations of S_n.
    November 8: Mio Alter - The Frobenius formula.
    November 15: Arun Chandrasekhar - Representations of A_n.
    November 20: Emily Tvetenstrand - Representations over the real numbers.
    November 29: Mio Alter - Representations of GL_2(F_q).
    December 6: Arun Chandrasekhar - Representations of GL_2(F_q), Part II.
    Homework 2 (Due October 18): Homework_2.pdf
    Homework 3 (Due November 8): Homework_3.pdf
    Homework 4 (Due November 29): Homework_4.pdf