Introduction to Grassmann and Schubert varieties, and their applications

Sudhir R. Ghorpade, Indian Institute of Technology Bombay (Mumbai, India)

Abstract: In these lectures, we will attempt to provide a gentle introduction to Grassmann varieties and its Schubert subvarieties, following a concrete approach. Applications to Coding Theory will also be outlined.

Lecture 1: Geometry of subspaces of a vector space

In this introductory lecture, we shall see how the collection of subspaces of a fixed dimension of a finite dimensional vector space has the geometric structure of a projective algebraic variety. This, then, is the Grassmann variety, and we will establish some of its basic properties. A cellular decomposition of Grassmann variety will be described and Schubert varieties in Grassmannians will be introduced.

Lecture 2: Standard Monomials and a Postulation Formula

We will give a concrete description of the homogeneous coordinate ring of a Grassmann variety and more generally, a Schubert variety in a Grassmannian, using “standard monomials” in Plücker coordinates. This will then be used to establish a postulation formula, due to Hodge, for Schubert varieties in Grassmannians. In other words, we give explicitly the Hilbert function as well as the Hilbert polynomials of Schubert varieties in Grassmannians. A combinatorial proof of this will be outlined.

Lecture 3: Applications of Grassmann and Schubert Varieties

We consider Grassmann and Schubert varieties over finite fields, and indicate how these can be used to construct interesting classes of linear error correcting codes. Some of the properties of these codes will be outlined, and some open problems may be mentioned.

Lecture 4: Extensions and Generalizations

If there is time and interest, we can discuss diverse topics related to the earlier lectures, such as, for instance, notion of a Schubert variety in more general set-up of suitable quotients of algebraic groups, the example of a Grassmann variety, and more generally, a partial flag variety, as an illustration of Weil conjectures, with a brief introduction to the latter, variants of linear codes related to Grassmann and Schubert varieties, and some recent results and questions concerning them.

Prerequisites: Linear algebra and some basics of abstract algebra and commutative algebra. Familiarity with rudiments of algebraic geometry and topology is desirable, but not essential.

Расписание лекций см. выше

#DLPM