6. A study of Equivariant Schubert Calculus | NYU Shanghai Undergraduate Research Symposium

audio presentation.mp3

Project Description

There are three parts in the report. First, we study Schur polynomials, a special kind of polynomials which are closely related to the representation of GL(n, C) and Schubert Calculus. In fact, the Schur polynomials turn out to be the algebraic foundation of Schubert Calculus. We shall use combinatoric approaches to study the algebraic properties of Schur polynomials, their alternative expression and the products of Schur polynomials (Pieri’s rule, Giambelli’s formula and Littlewood-Richardson’s rule), for instance.

The second part is the classical Schubert Calculus. It concerns the structure of Grassmannians Gr(k, n), the moduli spaces consists of all k-dimensional subspaces of the vector space C^n. We study the topological properties and algebraic structures of Schubert cells and Schubert classes, the elemental parts of Schubert Calculus. As a topological space, the Schubert cells of a given Grassmannian form a CW-Complex; on the other hand, the Schubert classes of a Grassmannian turns out to be a cohomology ring. In the later parts of this section, we study the properties of such cohomology rings, such as their multiplication rules, and their alternative expressions as quotient rings.

The last section is about the Equivariant Schubert Calculus, an extension to the traditional version of Schubert Calculus based on the concept of equivariant cohomology. To be more precise, it concerns the group action of torsions on a Grassmannian. We first introduce the (extended) definition of equivariant cohomology as the background knowledge. After that, we select a specific equivariant cohomology ring of Grassmannian, and study its multiplication products with the help of the computer programs.

Comments

I really like the problem of Graph 1, can you say more about it?

The question is stated just like below the graph, and the answer to it can be seen on the graph itself. For your question, I think I can supplement with some concepts in the Schubert Calculus, to illustrate how we deal with these problems in an algebraic manner.

As you know, the main object we study is the Grassmannian, and it is defined to be the set of all subspaces (of a fixed dimension) of a complex vector space. In Schubert Calculus, there is a kind of object called the Schubert Varieties (like I mentioned in the presentation), which is in fact a kind of subset of the Grassmannian that meets some special requirements. Though these requirements are defined in a very abstract way, they can actually be interpreted as the dimension of the intersections of these subspaces (in the Schubert varieties) with some given subspaces.

Under some special cases, the subspaces contained in the Grassmannian can be regarded as a special kind of “lines”, called the projective lines. In this way, the intersection requirements (under these cases) for the Schubert varieties can be interpreted as how do the “lines” contained in the Schubert varieties intersect with some given “lines” or even “planes”. This gives us a bridge from the geometric question to the algebraic solution to the question.

Hope this answer will give you more insights into the question! (Sorry for the late reply due to the busy final week!)