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

Hermann Schubert

Hermann Cäsar Hannibal Schubert was a German mathematician.Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen...

.

Background

described the history of Schubert polynomials.

The Schubert polynomials 𝔖_w are polynomials in the variables x₁, x₂,.... depending on an element w of the infinite symmetric group S_∞ of all permutations of 1, 2, 3,... fixing all but a finite number of elements. They form a basis for the polynomial ring Z[x₁, x₂,....] in infinitely many variables.

The cohomology of the flag manifold Fl(m) is Z[x₁, x₂,....x_m]/I, where I is the ideal generated by homogeneous symmetric functions of positive degree.
The Schubert polynomial 𝔖_w is the unique homogeneous polynomial of degree ℓ(w) representing the Schubert cycle of w in the cohomology of the flag manifold Fl(m) for all sufficiently large m.

Properties

• If w is the permutation of longest length in S_n then 𝔖_w = xx...x
• ∂_i𝔖_w = 𝔖_wsi if w(i)>w(i+1), where s_i is the transposition (i,i+1) and where ∂_i is the divided difference operator taking P to (P−s_iP)/(x_i−x_i+1).

Schubert polynomials can be calculated recursively from these two properties.

• 𝔖₁ = 1
• If w is the transposition (n,n+1) then 𝔖_w = x₁+...+x_n
• If w(i)<w(i+1) for all i≠r then 𝔖_w is the Schur polynomial s_λ(x₁,...,x_r) where λ is the partition (w(r)−r....,w(2)−2, w(1)−1). In particular all Schur polynomials (of a finite number of variables) are Schubert polynomials.

Double Schubert polynomials

Double Schubert polynomials 𝔖_w(x₁,x₂, ...y₁,y₂,...) 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 y_i are 0.

The double Schubert polynomial 𝔖_w(x₁,x₂, ...y₁,y₂,...) are characterized by the properties

• 𝔖_w(x₁,x₂, ...y₁,y₂,...) = Π_i+j≤n(x_i−y_j) when w is the permutation on 1,...,n of longest length.
• ∂_i𝔖_w = 𝔖_wsi if w(i)>w(i+1)

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.

See also

• Kostant polynomial
  Kostant polynomial
  In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system.-Background:...
  
  
• Monk's formula
  Monk's formula
  In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold.-References:...
  
   gives the product of a linear Schubert polynomial and a Schubert polynomial.
• nil-Coxeter algebra
  Nil-Coxeter algebra
  In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.-Definition:...