In this chapter we discuss the properties of Grassmannians and flag varieties. They can be defined as homogeneous spaces G/P for the general linear group G and a parabolic subgroup P, but we take a direct approach.

In section 3.1 we define the Plücker embedding and prove the quadratic relations satisfied by the image of the flag variety by the Plücker embedding in the product of projective spaces. They turn out to be the shuffling relations defining the Schur functors. This leads to the identification of the multihomogeneous components of the coordinate rings of flag varieties and Schur modules.

In section 3.2 we describe the standard coverings of flag varieties by affine spaces. We apply these coverings to prove the Cauchy formula stated in section 2.2.

In section 3.3 we define tautological vector bundles on flag varieties and define the flag varieties in a relative setting. We also prove an important result about a Koszul complex resolving the coordinate sheaf of tautological bundle on a Grassmannian. This is a prototype of the technique we will develop in the later chapters.

The Plücker Embeddings

Let E be a vector space of dimension n over a field K, and let r be an integer, 0 < r ≤ n.