Grassmannians and Flag Varieties
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Grassmannians and Flag Varieties book
This chapter introduces a building point of view for describing Grassmann varieties of kr+1. They index the canonical affine open sets of Grassmann varieties and allow a geometrical interpretation of their natural coordinates. Generally the set of combinatorial flags, corresponding to the flags of kr+1, adapted to the canonical basis, indexes the affine canonical open sets of flags varieties. The natural fibering in Grassmannians of flag varieties is obtained. Their projective variety structure over k is shown in terms of the Plucker and Segre's embeddings. A Grassmann variety may be canonically embedded in a projective space as a closed subvariety thus proving that it is a projective variety. Flag varieties are natural generalizations of Grassmannians. A chain of a finite set is informally speaking a redundant flag, given by an ascending sequence of subsets, with the inclusion relations not necessarily strict.