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 are polynomials in the variables depending on an element of the infinite symmetric group of all permutations of fixing all but a finite number of elements. They form a basis for the polynomial ring in infinitely many variables.
The cohomology of the flag manifold is where 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

Properties

Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that.
Other properties are
As an example

Multiplicative structure constants

Since the Schubert polynomials form a basis, there are unique coefficients
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 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 are.
The double Schubert polynomial 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 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.