18.217    Combinatorial Theory: Coxeter Combinatorics

Class meets: MWF 1-2 pm   Room 4-145        

Instructor: Alexander Postnikov

Synopsis:

We will discuss combinatorics of Coxeter groups, root systems, and related structures. Root systems are one of the central objects in Lie theory. They are classified by Dynkin diagrams (the Cartan-Killing classification). Remarkably, the same Dynkin diagrams appear in many other different contexts, e.g., in the classification of Fomin-Zelevinsky's cluster algebras of finite type. Many popular objects in combinatorics, such as Young tableaux and Schur polynomials, originally came from representation theory. We'll talk about generalizations of these and many other classical combinatorial objects in the Coxeter context. In particular, we'll discuss Coxeter-Catalan combinatorics. Some prior knowledge of Lie theory would be helpful; although it is not required.

Topics: (tentative list)

• Bert Kostant's game. Chip-firing game and other games on graphs. Vinberg's subadditive functions.

• Root systems, reflections, 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. Alcoved polytopes.

• Two Weyl's formulas: |W|=(e₁+1)...(e_r+1) and |W| = f r! a₁... a_r.

• Weyl's character and dimension formulas. Demazure's character formula.

• Coxeter-Catalan combinatorics. Catalan arrangements and root posets.

• Fomin-Zelevinsky's cluster algebras and generalized associahedra.

• Kostant's partition function. Flow polytopes.

• Artin groups and Hecke algebras. Connections with knot invariants.

• Cohomology and K-theory of generalized flag varieties G/B.

• Bernstein-Gelfand-Gelfand's divided differences and Kostant-Kumar's nil-Hecke algebra.

• Schubert polynomials, Kostant polynomials, and Grothendieck polynomials.

• Stembridge's fully-commutative elements of Coxeter groups.

• Pattern avoidance in root systems. Smooth Schubert varieties.

• Generalized Littlewood-Richardson coefficients. Berenstein-Zelevinsky's polytopes.

• Horn's problem, Klyachko cone, Knutson-Tao's honeycombs and hives,

• Kashiwara's crystal graphs, Littlemann's paths, alcove path model.

• Kazhdan-Lusztig polynomials.

Course Level: Graduate

Grading: The grade is based on several problem sets, in-class quizzes, and presentations.

Problem Sets: TBA

Lectures: 18.217 Lecture Notes by Ilani Axelrod-Freed

• Anders Bjorner, Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer, 2005.

• Nicolas Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 4-6, Springer, 2008.

• Sergey Fomin, Nathan Reading, Root Systems and Generalized Associahedra, IAS/Park City Mathematics Series, Vol. 14, 2004.

• 18.218 Lecture Notes, taught by A. Postnikov, notes taken by Evan Chen, Spring 2017.

Related past course: 18.217: Topics in Combinatorics: Coxeter Combinatorics, A. Postnikov, Spring 2017.

last updated: September 2, 2025