18.218 M.I.T. Spring 2017
18.218 Topics in Combinatorics:
MWF 2-3 pm
First class: Wednesday, February 8
apost at math dot mit dot edu
All kinds of combinatorial things around Coxeter groups and root systems ...
Roots systems are one of the central objects in Lie theory.
They have a very rich combinatorial structure. Root systems can be classified by Dynkin diagrams
(the Cartan-Killing classification). Remarkably, the same
ubiquitous Dynkin diagrams appear in many other different areas of mathematics (e.g.,
Arnold's classification of singularities,
Gabriel's theorem on quivers,
Fomin-Zelevinsky's cluster algebras of finite type, etc.)
The course will focus on combinatorics of root systems, Weyl groups,
and related structures.
Many popular objects in combinatorics, such as Young tableaux and
Schur symmetric polynomials, originally came
from representation theory of general linear and symmetric groups. They have more general Lie-theoretic analogs
(Weyl's characters, etc.) that we'll discuss in the course.
We'll talk about Lie-theoretic versions of the hook-length formula, Littlewood-Richardson rule, Robinson-Schensted-Knuth
correspondence, and other classical combinatorial constructions.
We'll talk about Kostant's partition function that has a simple combinatorial definition, but
surprisingly deep and non-trivial properties.
Root systems have beautiful enumerative features. Many famous integer sequences, such as the Catalan numbers and
the Fibonacci numbers, have root-theoretic generalizations, called
the Coxeter-Catalan numbers, etc.
We'll discuss convex polytopes and hyperplane arrangements related to root system:
generalized associahedra, alcoved polytopes,
affine Coxeter arrangements, etc.
We'll explain how to extend permutation pattern avoidance to root systems
and how this helps to describe smooth Schubert varieties.
And much much more ...
This course is not a replacement for a course on Lie theory.
Lie theory will serve as a motivation for us.
Our main focus will be put on combinatorics.
Some prior knowledge of representation theory and Lie theory would be helpful; although it is not required.
We'll discuss the following topics (tentative list, material will be covered as time allows):
- Bert Kostant's game and other games on graphs: "Find the highest root", chip-firing game, Cartan firing,
Vinberg's subadditive functions.
Root systems, reflections, roots, weights, Weyl groups, Dynkin diagrams, the Cartan-Killing classification.
- Reflection groups and Coxeter groups, reduced decompositions, the Bruhat order.
Coxeter elements, polynomial invariants, exponents.
Affine Weyl groups, affine Coxeter arrangements and alcoves. Minuscle weights.
Two Weyl's formulas for the order of the Weyl group:
|W|=(e1+1)...(er+1) = f r! a1... ar.
Weyl's character and dimension formulas.
Demazure's characters and Demazure's character formula.
Coxeter-Catalan combinatorics, the root poset, the Catalan arrangement, generalized associahedra.
Connections with cluster algebras.
Kostant's partition function. Its piecewise-polynomial properties. Flow polytopes. Connections with box-splines.
Cohomology and K-theory of flag varieties G/B:
Bernstein-Gelfand-Gelfand's divided differences and
Kostant-Kumar's nil-Hecke algebra.
Schubert polynomials, Kostant polynomials, and Grothendieck polynomials.
Pattern avoidance in root systems, smoothness of Schubert varieties,
Stembridge's fully-commutative elements of Coxeter groups.
Towards generalized Littlewood-Richardson:
Horn's problem, Klyachko cone, Knutson-Tao's honeycombs and hives,
Kashiwara's crystal graphs, Littlemann's paths, alcove path model.
- Kazhdan-Lusztig polynomials.
Affine Grassmannian and Lam's affine Stanley symmetric functions.
Course Level: Graduate
The course should be accessible to first year graduate students.
Grading: The grade will be based on several problem sets.
Problem Set 1: Due Monday, April 3, 2017
Optional Problem Set 2: Turn in by Wednesday May 17, 2017
Lecture Notes by Evan Chen
(the file will be updated periodically)
- W 02/08/2017. Bert Kostant's game and other games on graphs.
Allen Knutson's post
- F 02/10/2017. Chip-firing and Cartan firing. Finiteness and Uniqueness.
Bjorner, Lovasz, Shor:
Chip-firing games on graphs
Abelian sandpile model wiki
M 02/13/2017. No class - Snow day
- W 02/15/2017.
Proof of uniqueness: Diamond lemma and its generalization
- F 02/17/2017.
Vinberg's additive and subadditive functions. Finiteness.
Happel, Preiser, Ringel:
Vinberg's characterization of Dynkin diagrams using
subadditive functions with application to DTr-periodic modules.
M 02/21/2017. No classes on Monday.
(Monday schedule of classes on Tuesday)
- T 02/21/2017.
Generalized Cartan matrices.
- W 02/22/2017.
Farkas' lemma. Vinberg's theorem
[Kac, Theorem 4.3].
- F 02/24/2017.
Sketch of proof of Vinberg's theorem.
Its application to chip-firing.
The classification of generalized Cartan matrices of finite and affine types.
- M 02/27/2017.
The classification (cont'd).
- W 03/01/2017.
Reflections. Root systems. Weyl group. Positive and negative roots.
- F 03/03/2017.
Properties of simple reflections.
- M 03/06/2017. Decompositions of Weyl group elements
and walks on Weyl chambers.
- W 03/08/2017. Reduced decompositions. Inversions.
Statistics on permutations and their Weyl group analogs.
- F 03/10/2017. The root poset. The highest root.
The Coxeter number. The exponents. The index of connection.
Two formulas for the order of the Weyl group.
- M 03/13/2017. The root lattice and the weight lattice.
The affine Weyl group. The affine Coxeter arrangement.
- W 03/15/2017. The fundamental alcove. Alcoves
are simplices. Alcove walks. Proof of Weyl's formula.
- F 03/17/2017. Affine permutations. Wiring diagrams
and affine wiring diagrams. Centers of alcoves.
The strong Bruhat order and the weak Bruhat order.
Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 4-6, Springer, 2008.
[Bjoner-Brenti] Anders Bjorner, Francesco Brenti,
Combinatorics of Coxeter Groups,
Graduate Texts in Mathematics, 231, Springer, 2005.
- [Fomin-Reading] Sergey Fomin, Nathan Reading,
Root Systems and Generalized Associahedra,
IAS/Park City Mathematics Series, Vol. 14, 2004.
last updated: February 10, 2017