Math 6150F (Spring 2017)
Welcome to the homepage for Math 6150F: Coxeter systems and Iwahori-Hecke algebras!
- Send questions to emarberg@ust.hk.
For information about prerequisites, grades, and the course schedule, consult the syllabus.
Lectures: MW from 3:00PM to 4:20PM in Room 5506, Lift 25-26
Office hours: TTh from 3PM to 4PM (or by appointment) in Room 3492 in Math Department, Lift 25-26
Textbooks:
- Reflection Groups and Coxeter Groups by Humphreys
- Combinatorics of Coxeter Groups by Bjorner and Brenti
- Characters of Finite Coxeter Groups by Geck and Pfeiffer
All of these are good references. We will mainly follow Humphreys's book.
Lectures:
- Lecture 1:
Course details, the symmetric group as a Coxeter group
- Lecture 2:
Definitions of reflection groups and root systems
- Lecture 3:
Simple systems, examples, generation by simple reflections
- Lecture 4:
Length function and exchange principle for reflection groups
- Lecture 5:
Longest element, finite reflection groups are Coxeter groups
- Lecture 6:
Coxeter graphs, parabolic subgroups
- Lecture 7:
Coxeter groups in general, geometric representation
- Lecture 8:
Root systems, faithfulness of geometric representation
- Lecture 9:
Geometric length function, strong exchange principle
- Lecture 10:
Bruhat order and reduced expressions
- Lecture 11:
Grading of Bruhat order, coset representatives, Tits cone
- Lecture 12:
Positive definite implies finite
- Lecture 13:
Finite implies positive definite, crystallographic groups
- Lecture 14:
Classification of finite Coxeter groups
- Lecture 15:
Existence of generic Hecke algebras, examples
- Lecture 16:
Presentiation for generic Hecke algebra
- Lecture 17:
Hecke algebras as endomorphism algebras
- Lecture 18:
Examples from general linear groups, Iwahori-Hecke algebras
- Lecture 19:
Bar involution and Kazhdan-Lusztig basis
- Lecture 20:
Kazhdan-Lusztig polynomials, definition of cells
- Lecture 21:
Course recap, left cell representations
- Lecture 22:
Left cells in the symmetric group, RSK correspondence
- Lecture 23:
Split Grothendieck groups, categories of bimodules
- Lecture 24:
Soergel bimodules, categorification theorems
Assignments: