Schubert Varieties
Spring, 2005
Prof. Sara Billey
Wednesday, Friday 12:30-1:50 Loew
118
Syllabus
Summary:This course will introduce Schubert Varieties from a combinatorial
point of view. The course will build on the course taught this past
fall by Monty McGovern, however, it is possible to take this class
without having seen the first course. The necessary prerequisites are
just Abstract Algebra and basic Combinatorics (graphs, partitions,
compositions, posets, etc). The main topics we will cover include:
- Schubert Varieties in the Grassmannian Manifold
- Schubert Varieties in the Flag Manifold
- Bruhat order
- Pl\"ucker coordinates
- Schubert polynomials and the cohomology ring of the Grassmannian Manifold and the Flag Manifold
- Various rules for computing the cup product of these cohomology rings.
- Smoothness of Schubert varieties and their tangent spaces
- Revisiting Kazhdan-Lusztig polynomials.
- Degree of a Schubert variety
- Multiplicy of a point on a Schubert variety.
- Vanishing theorems.
- Connections to standard monomial theory and Littlemann Paths.
- Schubert theory for other semisimple Lie groups, Kac-Moody groups, and GKM spaces (as time permits).
Reference Texts:
- "Flag Varieties" by Gonciulea and
Lakshmibai. Actualites Mathematiques, 2001.
- "Symmetric
Functions, Schubert Polynomials and Degeneracy Loci" by Laurent
Manivel. American Mathematical Society and Societe Mathematique de
France, SMF/AMS Texts and Monographs, vol. 6, 2001.
- "Young Tableaux" by William Fulton. London Mathematical Society
Student Texts, vol. 35, 1997.
- "Kac-Moody Groups, Their Flag Varieties & Representation Theory"
by Shrawan Kumar. Birkhauser, Boston, Progress in Math Series, 2002.
- "Singular Loci of Schubert Varieties"
by Billey and Lakshmibai. Birkhauser, Boston, Progress in Math Series, 2000.
Research Papers: This course will cover material at the frontier of
research in algebraic geometry and combinatorics. Each student will
be expected to present one lecture on a research paper in this area.
Students can choose an article on their own subject to approval or
select one from those in the "class library". In order to make the
research papers as accessible as possible, the presenters will be
asked to write up lecture notes in advance for review by the
instructor and ultimatley handed out to the class. Students are
encouraged to work in pairs on the presentations.
Grading: There will be recommended exercises given in class but
these will not be officially graded. Grades will be based on the
student presentation and lecture notes.
Interesting Web Sites:
Last modified: Sun Mar 27 17:58:13 PST 2005