Schubert Varieties

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
• Multiplicity of a point on a Schubert variety.
• Vanishing theorems.
• Connections to standard monomial theory and Littelmann's Paths.
• Schubert theory for other semisimple Lie groups, Kac-Moody groups, and GKM spaces (as time permits).

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:

• Class Notes.
• Class Library: A collection of papers suitable for student presentations. Note, this is not a complete list of papers on this topic.
• Hermann Casar Hannibal Schubert 1848-1911.
• Hermann Grassmann 1809-1877.
• Combinatorics Seminar at UW
• GAP- Groups, Algorithms and Programming This is well written, reasonably fast, and free software for symbolic computation.
• Sloane's On-Line Encyclopedia of Integer Sequences
• JSTOR
• Graphviz Package Graph drawing software. Here is my dot file for Bruhat order. (Note, I haven't tried it with the latest version of Graphviz yet.)

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.

Tentative Schedule:

Wednesday Mar 28 -- Introduction to Schubert Varieties (notes by Soojin Cho)

Friday Mar 30 -- Equations defining Schubert varieties, rank conditions, and Bruhat order. (notes by Soojin Cho)

Wednesday Apr 4 -- Fulton's Essential Set. Using equations to determine smoothness of Schubert varieties. (notes by Brant Jones)

Friday Apr 6 -- Lie algebra and tangent spaces. (notes by Michael Goff)

Wednesday Apr 11 -- Tangent space bases for Schubert varieties and the maximal singular locus. (notes by Beth Kelly)

Friday Apr 13 -- Intersecting Schubert varieties. Grassmannians and other partial flag varieies. (notes by Kurt Luoto)

Wednesday Apr 18 -- Schubert polynomials, Schur functions and the cohomology ring of the Grassmannian Manifold and the Flag Manifold (notes Andrey Novoseltsev)

Friday Apr 20 -- Nice properties of Schubert polynomials using rc-graphs and other combinatorial gadgets. (notes by Ross Williams)

Wednesday Apr 25 -- Vakil's checkerboard game and permutations arrays. (notes by Luke Gutzwiller)

Friday Apr 27 -- Izzet Coskun lecturing on Mondrian Tableaux (notes by Sara Billey)

Wednesday May 2 -- Kostant polynomials, Kostant-Kumar theory and a bit about equivariant cohomology (notes by Steve Klee)

Friday May 4 -- Kumar's test for smoothness. Woo-Yong test for Gorenstein property. (notes by Anton Dochtermann)

Wednesday May 9 -- Kazhdan-Lusztig polynomials, small resolutions, and Bott-Samelson resolutions. (notes by Ashesh Bakshi)

Friday May 11 -- Lecture by Steve and Ashesh on "Flags, Schubert Polynomials, Degeneracy Loci, and Determinantal Formulas" by William Fulton.

Wednesday May 16 -- Lecture by Luke and Brant on paper by Woo and Yong (which one??)

Friday May 18 -- Lecture by Beth Kelly and Soojin Cho on "Skew Schubert Polynomials" by Lenart and Sottile.

Wednesday May 23 -- Lecture by Ross Williams and Michael Goff on "Combinatorics of Fulton's Essential Set" by Eriksson and Linusson.

Friday May 25 -- Lecture by Pete Littig

Wednesday May 30 -- Lecture by Steve Mitchell

Friday Jun 1 -- Lecture by Kurt and Andrey on "A Unified Approach to Combinatorial Formulas for Schubert Polynomials" by Cristian Lenart. Last modified: Fri Apr 13 12:05:41 PDT 2007