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
Instructor: Alexander Postnikov
apost at math
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
The course will include the following topics:
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
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
There are no any special prerequisites for the course.
Course Level: Graduate
Recommended (but not required) textbooks are:
||W. Fulton: Young Tableaux, Cambridge University Press, 1997.
||L. Manivel: Symmetric Functions,
Schubert Polynomials and Degeneracy Loci,
||R. P. Stanley: Enumerative Combinatorics,
Vol 2, Cambridge University Press, 1999.
- T 02/04/03. Course overview. Grassmannian: main definitions. [M, 3.1.1]
- R 02/06/03. Application: q-binomial coefficients. Plucker relations.
[F, 9.1], [M, 3.1.2].
- T 02/11/03. Schubert cells in the Grassmannian: 4 definitions.
[F 9.4], [M 3.2].
- 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]
- R 02/20/03. Cohomology ring the Grassmannian.
Duality theorem. [F, 9.4, Appendix B], [M, 3.2, Appendix].
- T 02/25/03. Cohomology ring the Grassmannian (cont'd).
[F, 9.4], [M, 3.2].
- R 02/27/03. Symmetric polynomials. Schur polynomials.
[EC2, 7.1-7.10, 7.15], [F, 6], [M, 1.1-1.2]
- T 03/04/03. Jacobi-Trudy identity and Giambelli formula.
[EC2, 7.9, 7.10, 7.16], [F, 6], [M, 1.2]
- R 03/06/03. Littlewood-Richardson rule: Classical rule and
[EC2, A1], [F, 5], [M 1.5].
- 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,
Problem Set 1 is due.
- R 03/13/03.
LR rule (cont'd): Klyachko cone, Berenstein-Zelevinsky polytope,
- T 03/18/03. Symmetric group, reduced decompositions, wiring diagrams,
weak Bruhat order. [M 2.1]
- 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]
- T 04/01/03. Schubert polynomials, divided differences, RC-graphs.
- R 04/03/03.
Schubert polynomials (cont'd): nilHecke algebra, Yang-Baxter equation,
Cauchy formula. [M 2.3, 2.4]
- T 04/08/03. Schubert polynomials (con'd): Grassmannian permutations,
relations to Schur polynomials, RC-graphs and families non-intersecting paths,
- R 04/10/03. Schubert polynomials (cont'd): Chevalley-Monk formula,
Pieri formula, Fomin-Kirillov quadratic algebra.
- T 04/15/03. Geometry of the flag manifold: Schubert cells and
varieties, cohomology ring, Chevalley-Monk formula (geometric variant).
- R 04/17/03. Borel theorem, coinvariant algebra.
T 04/22/03. [no class, Patriots day]
- R 04/24/03. Grobner bases. Calculating the generalized
LR-coefficients for the flag manifold.
- T 04/29/03. Guest lecture by Rom Pinchasi.
- R 05/01/03. Verma's theorem.
- T 05/06/03.
- R 05/08/03.
- T 05/13/03.
- R 05/15/03.