# 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 Lascoux & Schützenberger (1982) and are named after Hermann Schubert.

## Background

Lascoux (1995) described the history of Schubert polynomials.

The Schubert polynomials 𝔖w 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,....xm]/I, where I is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial 𝔖w 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.[citation needed]

## Properties

• If w is the permutation of longest length in Sn then 𝔖w = xn−1
1
xn−2
2
...x1
n-1
• i𝔖w = 𝔖wsi if w(i)>w(i+1), where si is the transposition (i,i+1) and where ∂i is the divided difference operator taking P to (PsiP)/(xixi+1).

Schubert polynomials can be calculated recursively from these two properties.

• 𝔖1 = 1
• If w is the transposition (n,n+1) then 𝔖w = x1+...+xn
• If w(i)<w(i+1) for all ir then 𝔖w is the Schur polynomial sλ(x1,...,xr) 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.
• 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.

## Double Schubert polynomials

Double Schubert polynomials 𝔖w(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 yi are 0.

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

• 𝔖w(x1,x2, ...y1,y2,...) = Πi+jn(xiyj) when w is the permutation on 1,...,n of longest length.
• i𝔖w = 𝔖wsi if w(i)>w(i+1)

## Quantum Schubert polynomials

Fomin, Gelfand & Postnikov (1997) 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

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