Schubert Variety - Significance

Significance

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.

The algebras of regular functions on Schubert varieties have deep significance in algebraic combinatorics and are examples of algebras with a straightening law. (Co)homology of the Grassmanian, and more generally, of more general flag varieties, is spanned by the (co)homology classes of Schubert varieties, the Schubert cycles. The study of the intersection theory on the Grassmanian was initiated by Hermann Schubert and continued by Zeuthen in 19th century under the heading of enumerative geometry. This area was deemed by David Hilbert important enough to be included as the fifteenth of his celebrated 23 problems. The study continued in 20th century as part of the general development of algebraic topology and representation theory, but accelerated in the 1990s beginning with the work of William Fulton on the degeneracy loci and Schubert polynomials, following up on earlier investigations of Bernstein–Gelfand–Gelfand and Demazure in representation theory in the 1970s, Lascoux and Schützenberger in combinatorics in the 1980s and of Fulton and MacPherson in intersection theory of singular algebraic varieties, also in the 1980s.