# 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: