Schubert Variety

In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Described by means of linear algebra, a typical example consists of the k-dimensional subspaces V of an n dimensional vector space W, such that

for j = 1, 2, ..., k, where

is a certain flag of subspaces in W and 0 < a₁ < ... < a_k ≤ n. More generally, given a semisimple algebraic group G with a Borel subgroup B and a standard parabolic subgroup P, it is known that the homogeneous space X = G/P, which is an example of a flag variety, consists of finitely many B-orbits that may be parametrized by certain elements of the Weyl group W. The closure of the B-orbit associated to an element w of the Weyl group is denoted by X_w and is called a Schubert variety in G/P. The classical case corresponds to G = SL_n and P being the kth maximal parabolic subgroup of G.