18.318   M.I.T.   Spring 2003

Topics in Combinatorics:   ``Schubert Calculus and Combinatorics''

"Problem 15: To establish a rigorous foundation of Schubert's enumerative calculus."
Mathematische Probleme by David Hilbert

Class meets: Tuesday, Thursday   11-12:30   room 2-102

Instructor: Alexander Postnikov   apost at math   room 2-389

Course webpage: http://www-math.mit.edu/~apost/courses/18.318/


The course is devoted to combinatorial aspects of Schubert calculus of the Grassmannian and the flag manifold. It is a classical area of enumerative geometry whose purpose is to calculate various intersection numbers and solve problems like the following:

Find the number of lines in the 3-dimensional complex space that intersect with given four generic lines.

Schubert calculus has links with combinatorics of symmetric functions and representation theory of the general linear group.

The course will include the following topics: Grassmann-Plucker relations, Schubert cells, Pieri's formula, Young tableaux, Schur symmetric polynomials, Jacobi-Trudy and Giambelli's formulas, Littlewood-Richardson rule, Gelfand-Serganova cells and matroids, Bruhat order, Chevalley-Monk's formula, Schubert polynomials, Bernstein-Gelfand-Gelfand-Demazure theorem, Cauchy formula, RC-graphs, etc. We will also discuss some recent results related to quantum cohomology and Gromov-Witten invariants, total positivity and links with inverse boundary problem for planar networks. Preference will be given to explicit combinatorial constructions and proofs.

The course will be self-contained. All required notions and definitions will be given. There are no any special prerequisites for the course.

Course Level: Graduate

Texts: Recommended (but not required) textbooks are:

[F]   W. Fulton: Young Tableaux, Cambridge University Press, 1997.
[M]   L. Manivel: Symmetric Functions, Schubert Polynomials and Degeneracy Loci, AMS, 2001.
[EC2]   R. P. Stanley: Enumerative Combinatorics, Vol 2, Cambridge University Press, 1999.


  1. T 02/04/03. Course overview. Grassmannian: main definitions. [M, 3.1.1]

  2. R 02/06/03. Application: q-binomial coefficients. Plucker relations. [F, 9.1], [M, 3.1.2].

  3. T 02/11/03. Schubert cells in the Grassmannian: 4 definitions. Schubert varieties. [F 9.4], [M 3.2].

  4. R 02/13/03. Matroids. Matroid stratification of the Grassmannian.
    [Gelfand, Goresky, MacPherson, Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math. 63 (1987) 301-316.]

    T 02/18/03. [no class, Monday schedule]

  5. R 02/20/03. Cohomology ring the Grassmannian. Duality theorem. [F, 9.4, Appendix B], [M, 3.2, Appendix].

  6. T 02/25/03. Cohomology ring the Grassmannian (cont'd). Pieri's formula. [F, 9.4], [M, 3.2].

  7. R 02/27/03. Symmetric polynomials. Schur polynomials. [EC2, 7.1-7.10, 7.15], [F, 6], [M, 1.1-1.2]

  8. T 03/04/03. Jacobi-Trudy identity and Giambelli formula. [EC2, 7.9, 7.10, 7.16], [F, 6], [M, 1.2]

  9. R 03/06/03. Littlewood-Richardson rule: Classical rule and Zelevinsky's pictures. [EC2, A1], [F, 5], [M 1.5].

  10. T 03/11/03. LR rule (cont'd): Berenstein-Zelevinsky triangles, Knutson-Tao honeycombs. [EC2, A1], [Knutson, Tao, The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, math.RT/9807160].
    Problem Set 1 is due.

  11. R 03/13/03. LR rule (cont'd): Klyachko cone, Berenstein-Zelevinsky polytope, Gelfand-Tsetlin polytope.

  12. T 03/18/03. Symmetric group, reduced decompositions, wiring diagrams, weak Bruhat order. [M 2.1]

  13. R 03/20/03. Strong Bruhat order. [M 2.1]

    T 03/25/03. [no class, Spring break]

    R 03/27/03. [no class, Spring break]

  14. T 04/01/03. Schubert polynomials, divided differences, RC-graphs. [M 2.3]

  15. R 04/03/03. Schubert polynomials (cont'd): nilHecke algebra, Yang-Baxter equation, Cauchy formula. [M 2.3, 2.4]

  16. T 04/08/03. Schubert polynomials (con'd): Grassmannian permutations, relations to Schur polynomials, RC-graphs and families non-intersecting paths, Gessel-Viennot method.

  17. R 04/10/03. Schubert polynomials (cont'd): Chevalley-Monk formula, Pieri formula, Fomin-Kirillov quadratic algebra.

  18. T 04/15/03. Geometry of the flag manifold: Schubert cells and varieties, cohomology ring, Chevalley-Monk formula (geometric variant).

  19. R 04/17/03. Borel theorem, coinvariant algebra. Grobner bases.

    T 04/22/03. [no class, Patriots day]

  20. R 04/24/03. Grobner bases. Calculating the generalized LR-coefficients for the flag manifold.

  21. T 04/29/03. Guest lecture by Rom Pinchasi.

  22. R 05/01/03. Verma's theorem.

  23. T 05/06/03.

  24. R 05/08/03.

  25. T 05/13/03.

  26. R 05/15/03.