Schubert variety

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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

\dim(V\cap W_{j})\geq j

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

W_{1}\subset W_{2}\subset \cdots \subset W_{k},\quad \dim W_{j}=a_{j}

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.

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 Grassmannian, 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 Grassmannian 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.

See also

References

  • P.A. Griffiths, J.E. Harris, Principles of algebraic geometry, Wiley (Interscience) (1978)
  • A.L. Onishchik (2001), "Schubert variety", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  • H. Schubert, Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension Mitt. Math. Gesellschaft Hamburg, 1 (1889) pp. 134–155
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