Grassmannians, Schubert Varieties & Standard Monomials

In this talk, we will show that the Grassmannian $\mathbb{G}(k,n)$ can be embedded into projective space as a projective variety via the Plücker embedding. Using standard monomial theory, we will describe a basis for the homogeneous coordinate ring of $\mathbb{G}(k,n)$. Finally, we will introduce the Schubert subvarieties of $\mathbb{G}(k,n)$. These arise from a stratification of $\mathbb{G}(k,n)$ by row reduced echelon form type of its elements (viewed as $(k+1)\times (n+1)$ matrices). Necessary prerequisite material will be covered at the start of the talk.