Schubert Calculus

Course guide:

Content:

Schubert calculus began with the work of Schubert, Pieri, and Giambelli in the 19th century, which studied the geometry of the Grassmannian manifold parametrizing k-dimensional linear subspaces of a complex vector space. This research was a precursor to the cohomology rings of more general manifolds and the Chern classes of vector bundles over them developed in the 20th century. Around the same time, important connections between Schubert calculus and the much older theory of symmetric functions and Schur polynomials were discovered.

Thanks to the pioneering papers of Borel and Chevalley in the 1950s, the subject today may be defined as the study of the cohomology ring of the compact homogeneous spaces G/P of complex Lie groups. Despite its long history, for the basic examples of symplectic and orthogonal Grassmannians, the analogues of the Schur polynomials were found only during the last 15 years. Moreover, a new, intrinsic point of view in Schubert calculus has emerged, which includes all of the classical groups, and employs algebro-combinatorial tools such as partitions, raising operators, Young tableaux, divided differences, nilCoxeter algebras, and transition trees. This has solved open problems in the field and revealed links with other areas such as the theory of Hall-Littlewood functions and Macdonald polynomials.

This course will be an introduction to Schubert calculus, with emphasis on the modern intrinsic perspective. I will strive to keep the discussion as self-contained as possible, so will NOT assume familiarity with algebraic geometry, intersection theory, characteristic classes, etc. However, a basic knowledge of Lie groups and reflection groups should be helpful. Although the geometry will remain mostly in the background, it is the motivation for much of the rich algebraic combinatorics that is encountered in the class. Understanding how these two areas interact with the Lie theory is one of the end goals of this subject.

Homework:

I plan to distribute some homework problems during the course. The homework will include material that could not be part of the lectures for lack of time.

The most relevant references are my two research-expository papers below. They give an overview of the theory with examples, but include very few proofs.

• Giambelli and degeneracy locus formulas for classical G/P spaces, pdf
  Mosc. Math. J. 16 (2016), 125-177.

• Theta and eta polynomials in geometry, Lie theory, and combinatorics, pdf
  "First Congress of Greek Mathematicians", 243-284,
  De Gruyter Proc. Math., De Gruyter, Berlin, 2020.

Some older references in book form:

• H. Hiller, "Geometry of Coxeter Groups", Research Notes in Mathematics 54, Pitman, Boston 1982.

• W. Fulton, "Young Tableaux, with Applications to Representation Theory and Geometry", Cambridge Univ. Press, Cambridge 1997.

• L. Manivel, "Symmetric Functions, Schubert Polynomials and Degeneracy Loci", Amer. Math. Soc., 2001.

• S. Kumar, "Kac-Moody groups, their Flag Varieties and Representation Theory", Birkhaüser, Boston 2002.

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