18.318 Topics in Combinatorics

Course meets: Tuesdays and Thursdays, 1-2:30, Room 13-3101.
Lecturer: Sergey Fomin, Room 2-363B, 253-1713, fomin@math.mit.edu

Topic for spring 1998: Schubert calculus.

Hermann Cńsar Hannibal Schubert (1848-1911)

Course Outline
I. Combinatorics of Coxeter groups.
II. Schubert calculus on a Grassmannian.
III. Schubert polynomials (classical case).
IV. Variations on the theme of Schubert.

Texts. The course will not strictly follow a particular text. Principal sources:
W.Fulton, Young tableaux , Cambridge University Press, 1997 (recommended text).
H.Hiller, Geometry of Coxeter groups , Pitman Publishing, 1982.
J.E.Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1994.
I.G.Macdonald, Notes on Schubert polynomials , UQAM, LACIM, 1990.
L.Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, SMF, 1998.

Lecture 1. Course overview. Finite reflection groups.
Lecture 2. Root systems.
Lecture 3. Deletion condition.
Lecture 4. Tits' theorem.
Lecture 5. Abstract Coxeter groups.
Lecture 6. Strong exchange property.
Lecture 7. Bruhat order.
Lecture 8. Verma's theorem.
Lecture 9. Weak order.
Lecture 10. Grassmannians. Plücker embedding.
Lectures 11-12. Schubert varieties and cells.
Lecture 13. Cohomology of algebraic varieties.
Lecture 14. Pieri's formula.
Lecture 15. Schur functions.
Lecture 16. Ring of symmetric functions.
Lecture 17. Cohomology ring of a Grassmannian.
Lecture 18. Schubert cells in flag manifolds.
Lecture 19. Intersections of Schubert varieties.
Lecture 20. Divided differences. Coinvariant algebra.
Lecture 21. NilCoxeter algebra. Line configurations.
Lecture 22. Combinatorial description of Schubert polynomials.
Lecture 23. Monk's formula.
Lecture 24. Cohomology of the flag manifold.
Lecture 25. Macdonald's identity. Stanley's polynomials.
Lecture 26. Enumeration of reduced words.

Problem Sets (enough to solve 3 problems in each)
Problem Set 1
Problem Set 2: Fulton, Exercises 17-21, page 152 (formula in ex.17 has to be corrected)
Problem Set 3