Schubert polynomial

In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Hermann Schubert.
Background.
 described the history of Schubert polynomials.
The Schubert polynomials formula_1 are polynomials in the variables formula_2 depending on an element formula_3 of the infinite symmetric group formula_4 of all permutations of formula_5 fixing all but a finite number of elements. They form a basis for the polynomial ring formula_6 in infinitely many variables.
The cohomology of the flag manifold formula_7 is formula_8 where formula_9 is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial formula_1 is the unique homogeneous polynomial of degree formula_11 representing the Schubert cycle of formula_12 in the cohomology of the flag manifold formula_7 for all sufficiently large formula_14
Properties.
Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that formula_25.
Other properties are
As an example 
Multiplicative structure constants.
Since the Schubert polynomials form a basis, there are unique coefficients formula_37 
such that 
These can be seen as a generalization of the Littlewood−Richardson coefficients described by the Littlewood–Richardson rule.
For representation-theoretical reasons, 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.
Double Schubert polynomials formula_39 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 formula_40 are formula_41.
The double Schubert polynomial formula_39 are characterized by the properties
The double Schubert polynomials can also be defined as 
Quantum Schubert polynomials.
 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.
Universal Schubert polynomials.
 introduced universal Schubert polynomials, that generalize classical and quantum Schubert polynomials. He also described universal double Schubert polynomials generalizing double Schubert polynomials.