Schubert Cycle
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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a Schubert variety is a certain
subvariety A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked: *below that of variety (''varietas'') *above that of form (''forma''). Subva ...
of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
, whose points correspond to certain kinds of subspaces ''V'', specified using linear algebra, inside a fixed vector subspace ''W''. Here ''W'' may be a vector space over an arbitrary field, though most commonly over the complex numbers. A typical example is the set ''X'' whose points correspond to those 2-dimensional subspaces ''V'' of a 4-dimensional vector space ''W'', such that ''V'' non-trivially intersects a fixed (reference) 2-dimensional subspace ''W''2: :X \ =\ \. Over the real number field, this can be pictured in usual ''xyz''-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of \mathbb(W), we obtain an open subset ''X''° ⊂ ''X''. This is isomorphic to the set of all lines ''L'' (not necessarily through the origin) which meet the ''x''-axis. Each such line ''L'' corresponds to a point of ''X''°, and continuously moving ''L'' in space (while keeping contact with the ''x''-axis) corresponds to a curve in ''X''°. Since there are three degrees of freedom in moving ''L'' (moving the point on the ''x''-axis, rotating, and tilting), ''X'' is a three-dimensional real algebraic variety. However, when ''L'' is equal to the ''x''-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes ''L'' a singular point of ''X''. More generally, a Schubert variety is defined by specifying the minimal dimension of intersection between a ''k''-dimensional ''V'' with each of the spaces in a fixed reference flag W_1\subset W_2\subset \cdots \subset W_n=W, where \dim W_j=j. (In the example above, this would mean requiring certain intersections of the line ''L'' with the ''x''-axis and the ''xy''-plane.) In even greater generality, given a semisimple algebraic group ''G'' with a Borel subgroup ''B'' and a standard parabolic subgroup ''P'', it is known that the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
''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 ''k''th 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 Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb ...
and are examples of algebras with a straightening law. (Co)homology of the Grassmannian, and more generally, of more general flag varieties, has a basis consisting of 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 the 19th century under the heading of enumerative geometry. This area was deemed by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
important enough to be included as the fifteenth of his celebrated 23 problems. The study continued in the 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 In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern c ...
and Schubert polynomials, following up on earlier investigations of
Bernstein Bernstein is a common surname in the German language, meaning "amber" (literally "burn stone"). The name is used by both Germans and Jews, although it is most common among people of Ashkenazi Jewish heritage. The German pronunciation is , but in E ...
Gelfand ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to: * People: ** Alan Gelfand, the inventor of the ollie, a skateboarding move ** Alan E. Gelfand, a statistician ** Boris Gelfand, a chess grandmaster ** Israel Gel ...
Gelfand ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to: * People: ** Alan Gelfand, the inventor of the ollie, a skateboarding move ** Alan E. Gelfand, a statistician ** Boris Gelfand, a chess grandmaster ** Israel Gel ...
and
Demazure Michel Demazure (; born 2 March 1937) is a French mathematician. He made contributions in the fields of abstract algebra, algebraic geometry, and computer vision, and participated in the Nicolas Bourbaki collective. He has also been president of ...
in representation theory in the 1970s, Lascoux and
Schützenberger Schützenberger may refer to these people: * Anne Ancelin Schützenberger (1919–2018) (de) * Paul Schützenberger, French chemist * René Schützenberger, French painter * Marcel-Paul "Marco" Schützenberger, French mathematician and Doctor of M ...
in combinatorics in the 1980s, and of Fulton and MacPherson in intersection theory of singular algebraic varieties, also in the 1980s.


See also

* Schubert calculus * Bruhat decomposition * Bott–Samelson resolution


References

*P.A. Griffiths, J.E. Harris, ''Principles of algebraic geometry'', Wiley (Interscience) (1978) * *H. Schubert, ''Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension'' Mitt. Math. Gesellschaft Hamburg, 1 (1889) pp. 134–155 {{Authority control Algebraic geometry Representation theory Commutative algebra Algebraic combinatorics