The Grassmannian and Its Schubert Varieties

In this chapter, we introduce Grassmannian varieties and their Schubert subvarieties; we have also included a brief introduction to flag varieties. We present the Standard Monomial Theory (abbreviated SMT) for Grassmannian and Schubert varieties. As a prime application of SMT, we present a proof of the vanishing of the higher cohomology groups of ample line bundles on the Grassmannian (and their restrictions to Schubert varieties).