18.217 Combinatorial Theory: Young tableaux

Fall 2022, 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:

The content of 18.217 varies from year to year.

This year we plan to concentrate on combinatorics of Young tableaux and their various generalizations. We will discuss combinatorial structures that appear in representation theory, geometry, and other areas.

We will talk about partitions, permutations, Young tableaux, Young's lattice, Gelfand-Tsetlin patterns, Bruhat orders, symmetric functions, representations of symmetric and general linear groups, Schur functions, Demazure characters, Schubert polynomials, Grassmannian and flag manifolds, pipe dreams, and (if time allows) crystal graphs, Littelmann path model, piecewise-linear combinatorics, tropical geometry, cluster algebras, total positivity, quantum cohomology, ...

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

Grading: Based on several problem sets.

Lectures: (with additional reading materials)

  1. W 9/7/2022: Introduction. Partitions. Young diagrams. Standard Young tableaux. Catalan numbers.
    18.212 lecture 2 from 2021 (pages 8, 9).

  2. F 9/9/2022: Hook length formula. Polytopal proof of the hook length formula. Piecewise-linear combinatorics. Toggles.
    [Sagan_SG, section 3.10]
    probabilistic proof: 18.212 lecture 4 from 2021 (pages 8-17).

  3. M 9/12/2022: Map φλ. Rectangular and diagonal sums. Robinson-Schensted correspondence.
    [Sagan_SG, Sections 3.1, 3.2, 3.3]

Lecture Notes taken by Ilani Axelrod-Freed: Lectures 1-4   Lectures 6-7   Lecture 9

Recommended books: (The students are not required to have these books. The material of the course has a nonempty intersection with the union of these three books. But we might present the material in a different order.)

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

last updated: September 9, 2022