18.715. INTRODUCTION TO REPRESENTATION THEORY
Tue-Thu 2:30-4, Room 4-149
Lecturer: George Lusztig
Graders: Enrico Colon, Gefei Dang
Office hour: Thu. 1-2pm (G.L. 2-365); Mon. 11am-12 (Gefei Dang 2-239).
There will be about 11 problem sets, assigned (almost) every Tuesday and due the following Tuesday by 11:59pm. You can submit homework in class or by email to a grader.
In early May you are supposed to submit a final project on a topic not covered in class.
The final grade will be based on problem sets (75/100), final project(25/100).
1) Representations of finite groups.
(First part of Serre's "Linear representations of finite groups".)
https://link.springer.com/book/10.1007%2F978-1-4684-9458-7.
2) Representations of the symmetric group (following Specht 1932).
3) Bruhat decomposition in GL_n, Hecke algebra.
4) Representations of GL_2 over a finite field (following Frobenius).
5) Representations of GL_n over a finite field appearing in functions on the flag manifold (following Steinberg's 1951 paper).
6) Modular representations of GL_n over a finite field, following Carter-Lusztig 1976.
7) Rational representations of GL_n (following Chevalley).
8) Hecke algebra and its new basis (following Kazhdan-Lusztig 1979 ).
9) Weyl character formula.
Homework 1, due Feb.14
Homework 2, due Feb.23
Homework 3, due Feb.28
Homework 4, due March 7
Homework 5, due March 14
Homework 6, due March 21
Homework 7, due Apr.4
Homework 8, due Apr.11
Homework 9, due Apr.18
Homework 10, due Apr.25
The final project is due Tuesday,May 9. It can be submitted on paper in class or sent as a pdf file to me. It should be an expository paper of 4 to 5 pages on a topic not covered in class. Example of possible topics: Artin's theorem (in Serre's book), Rationality properties of representations of finite groups (in Serre's book). You can choose another topic, but first asked me if I agree with it.