In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties.
They were introduced by Lascoux & Schützenberger (1982) and are named after Hermann Schubert.
Lascoux (1995) described the history of Schubert polynomials.
The Schubert polynomials
are polynomials in the variables
depending on an element
of the infinite symmetric group
fixing all but a finite number of elements.
They form a basis for the polynomial ring
The cohomology of the flag manifold
is the ideal generated by homogeneous symmetric functions of positive degree.
The Schubert polynomial
is the unique homogeneous polynomial of degree
representing the Schubert cycle of
in the cohomology of the flag manifold
for all sufficiently large
[citation needed] Schubert polynomials can be calculated recursively from these two properties.
Other properties are As an example Since the Schubert polynomials form a
-basis, there are unique coefficients
β γ
such that These can be seen as a generalization of the Littlewood−Richardson coefficients described by the Littlewood–Richardson rule.
For algebro-geometric reasons (Kleiman's transversality theorem of 1974), these coefficients are non-negative integers and it is an outstanding problem in representation theory and combinatorics to give a combinatorial rule for these numbers.
Double Schubert polynomials
are polynomials in two infinite sets of variables, parameterized by an element w of the infinite symmetric group, that becomes the usual Schubert polynomials when all the variables
The double Schubert polynomial
are characterized by the properties The double Schubert polynomials can also be defined as Fomin, Gelfand & Postnikov (1997) introduced quantum Schubert polynomials, that have the same relation to the (small) quantum cohomology of flag manifolds that ordinary Schubert polynomials have to the ordinary cohomology.
Fulton (1999) introduced universal Schubert polynomials, that generalize classical and quantum Schubert polynomials.
He also described universal double Schubert polynomials generalizing double Schubert polynomials.