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.

Background

edit

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   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  [citation needed]

Properties

edit
  • If   is the permutation of longest length in   then  
  •   if  , where   is the transposition   and where   is the divided difference operator taking   to  .

Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that  .

Other properties are

  •  
  • If   is the transposition  , then  .
  • If   for all  , then   is the Schur polynomial   where   is the partition  . In particular all Schur polynomials (of a finite number of variables) are Schubert polynomials.
  • Schubert polynomials have positive coefficients. A conjectural rule for their coefficients was put forth by Richard P. Stanley, and proven in two papers, one by Sergey Fomin and Stanley and one by Sara Billey, William Jockusch, and Stanley.
  • The Schubert polynomials can be seen as a generating function over certain combinatorial objects called pipe dreams or rc-graphs. These are in bijection with reduced Kogan faces, (introduced in the PhD thesis of Mikhail Kogan) which are special faces of the Gelfand-Tsetlin polytope.
  • Schubert polynomials also can be written as a weighted sum of objects called bumpless pipe dreams.

As an example

 

Multiplicative structure constants

edit

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 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

edit

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

  •   when   is the permutation on   of longest length.
  •   if  

The double Schubert polynomials can also be defined as

 

Quantum Schubert polynomials

edit

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.

Universal Schubert polynomials

edit

Fulton (1999) introduced universal Schubert polynomials, that generalize classical and quantum Schubert polynomials. He also described universal double Schubert polynomials generalizing double Schubert polynomials.

See also

edit

References

edit