18.217   Combinatorial Theory:
Schur polynomials and Schubert polynomials

Fall 2023, MIT

Instructor: Alex Postnikov
Grader: TBA    
Class meetings: MWF 1-2 pm     Room 56-114
Webpage: math.mit.edu/~apost/courses/18.217/

Course description:

Schur polynomials and Schubert polynomials play an important role in many different areas of mathematics (such as representation theory, algebraic geometry, etc). We will discuss beautiful properties of these polynomials, and their various generalizations, focusing on their combinatorial aspects. A large portion of current research in algebraic combinatorics is devoted to the study of these polynomials. We'll discuss some of these advances.

The course will include the following topics: combinatorics of partitions and permutations, Schur polynomials, Schubert polynomials, Demazure characters, Grothendieck polynomials, quantum Schubert polynomials, Young tablaux, RC-graphs (aka pipe dreams), RSK correspondence, Chevalley-Monk and Pieri formulas, Bernstein-Gelfand-Gelfand formula, Littlewood-Richardson rule and its various generalizations, Kashiwara's crystal graphs, alcove path model, etc.

Course Level: The course should be accessible to first year graduate students.

Grading: Based on several problem sets.

TBA

Lectures:

TBA

Some recommended books:

Here are some additional reading materials. The students are not required to read all these books. But is it a good idea to take a look at them. The course will cover some sections from them. The topics of the course are not limited to the union of these books. When needed, we will provide additional references to research papers.

• I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, pdf.
• Richard P. Stanley, Enumerative Combinatorics, Volume 2. Chapter 7.
• William Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, Series Number 35.
• Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, Providence, R.I.

This webpage will be updated periodically. All information related to the course, including problem sets, will be posted here.

last updated: September 5, 2023