Schubert cycle
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 vector 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 < ... < ak ≤ 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 Xw and is called a Schubert variety in G/P. The classical case corresponds to G = SLn and P being the kth maximal parabolic subgroup of G.
Schubert varieties form one of the most important and best studied classes of singular algebraic varieties. A certain measure of singularity of Schubert varieties is provided by Kazhdan–Lusztig polynomials, which encode their local Goresky–MacPherson intersection cohomology.
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