Math 6150I (Spring 2020)

Overview:

Welcome to the homepage for Math 6150I: Combinatorics of crystal bases!
Send questions to eric.marberg@gmail.com.
For information about prerequisites, grades, and the course schedule, consult the syllabus.

Lectures:

Wednesdays and Fridays from 1:30PM to 2:50PM
For the time being, lectures will be conducted online using zoom.
For information on how to join the online lectures, contact the instructor.

Office hours:

By appointment in Room 3492 in Math Department, Lift 25-26

Textbooks:

Crystal Bases: Representations and Combinatorics by Bump and Schilling
Introduction to Quantum Groups and Crystal Bases by Hong and Kang
We will mainly follow Bump and Schilling's book, so it would be good to obtain a copy.

Lectures:

Lecture 1: Course details, Schur polynomials, type A crystals
Lecture 2: Crystals of words and tableaux
Lecture 3: RSK correspondence
Lecture 4: Cartan types and Kashiwara crystals
Lecture 5: Properties of crystals for general Cartan types
Lecture 6: Stembridge crystals
Lecture 7: Virtual crystals
Lecture 8: Fundamental crystals
Lecture 9: Normal crystals for general Cartan types
Lecture 10: The plactic monoid and skew tableaux
Lecture 11: Representation theory of general linear groups
Lecture 12: Schur-Weyl duality
Lecture 13: Extensions of RSK and bicrystals
Lecture 14: Cauchy correspondence and Edelman-Greene insertion
Lecture 15: String patterns
Lecture 16: B infinity crystals
Lecture 17: Demazure crystals and characters
Lecture 18: The star involution
Lecture 19: More properties of the star involution
Lecture 20: Crystals and tropical geoemtry
Lecture 21: Lusztig parametrization of B infinity
Lecture 22: MV polytopes
Lecture 23: More on MV polytopes
Lecture 24: Further topics

Assignments:

Homework 1 is due Monday, March 9
Homework 2 is due Wednesday, March 18
Homework 3 is due Monday, March 30
Homework 4 is due Monday, April 20
Homework 5 is due Friday, May 8