Schubert varieties of Grassmannians

Abstract

In this endeavor, studying Grassmannians in a ne and projective space is essential. We rstly present Grassmannians as varieties then divide Grassmannians into disjoint Schubert cells and apply Plucker embedding to realize Grassmannian as a subvariety of projective space. We also brie y introduce the similar concepts for Flag manifolds. In this thesis, one of our aim is to expand the underlying ideas behind the Schubert calculus. For this purpose, we introduce some enumerative problems. We also review the Schubert polynomials and Schur functions. Finally, we associate the ring of symmetric polynomials with the cohomology ring of Grassmannians by indexing the Schubert class of cohomology ring of Grassmannians with Schur functions. To understand the product of Schur functions Pieri's formula is given. Moreover, we associate the products of Schur functions with the intersection of Schubert varieties of Grassmannians. Similarly, to understand the product of Schubert polynomials Monk's rule is given. Then the Schubert class of cohomology ring of Flag manifold is indexed by Schubert polynomials and we associate the products of Schubert polynomials with the intersection of Schubert varieties of Flag manifold.