| Call number: | 29298 | |
| Room/Time: | MW 2:40pm--3:55pm, 417 Math | |
| Instructor: | Mikhail Khovanov | |
| Office: | 620 Math | |
| Office hours: | Monday 4:15-5:15pm, Wednesday 1:30-2:30pm | |
| E-mail: | | |
| Teaching Assistant: | Darren Gooden | |
| TA email: | | |
| TA office hours: | Tuesday 10am-1pm in Help Room 406 Math | |
| Midterm : | Wednesday March 9 | |
| Review Session: | Thursday May 5, 4:00-5:30pm, 507 Math | |
| Final: | Wednesday May 11, 1:10-4pm, 417 Math | |
| Webpage: | www.math.columbia.edu/~khovanov/finite2016 | |
Prerequisites Both semesters of "Introduction to Modern Algebra" or equivalent are required. You need to be comfortable with the theory of groups, rings and fields. You also need to be know how to write proofs.
Textbook Representation Theory of Finite Groups, by Benjamin Steinberg Available via Amazon and other online merchants.
Syllabus:
Rings and modules over them. Idempotents. Simple modules.
Actions of groups on vector spaces, representations.
Group algebras, modules, complete reducibility.
Schur's lemma, irreducible modules, characters, orthogonality relations.
Fourier transform on finite groups.
Tensor products of representations. Symmetric and exterior powers of
a representation.
Restriction and induction operations. Frobenius reciprocity.
Representations of algebras and rings.
The McKay correspondence, Dynkin diagrams, reflection groups.
Representations of the symmetric group.
If time allows: Hopf algebras and their representations.
Homework: Homework
will be assigned on Wednesdays, due Wednesday the next week before class.
The first problem set is due January 27. The lowest homework score
will be dropped.
The numerical grade for the course will be the following linear combination:
35% homework, 25% midterm, 40% final.
Homework 1 Due January 27.
Homework 2 Due February 3.
Homework 3 Due February 10.
Homework 4 Due February 17.
Homework 5 Due Monday, February 29.
Homework 6 Due March 2.
Homework 7 Due March 9.
Homework 8 Due March 30.
Homework 9 Due April 6.
Homework 10 Due April 13.
Homework 11 Due April 27.
Online resources: See more at the end of the 2009 Representation of finite groups webpage.
Representations of Finite Groups Course Notes by Katherine Christianson. These are notes of our entire course.
Classification of finite subgroups of SO(3): by Hannah Mark and Maud De Visscher. Also discussed in M.Artin's Algebra textbook.
D.B.Westra in SU(2) and SO(3) gives more details about the surjection of SU(2) onto SO(3).
Qi You, M.K., Notes on the McKay correspondence
For our lecture 6 on Monday, February 8, read Chapter 2 (pages 14-18 of book, correspond to pages 21-25 of pdf) of Peter Webb's Representation Theory Book.
Modules over general rings are not discussed in the textbook. You can read about them in Chapter 1: Modules, sections 1.1-1.2 and 1.3-1.6 (up to page 12) or in Module fundamentals, first 8 pages.
A comprehensive treatment of modules can be found in Dummit and Foote, Modern Algebra, Sections 10.1-10.3. In addition, Chapters 18, 19 give excellent coverage of representations of finite groups. This book is currently on reserve at the Math library (3 copies).
Optional reading: Proof that every finite division ring is a field.