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 $\mathfrak{S}_w$ are polynomials in the variables $\ x_1,x_2,\ldots$ depending on an element $w$ of the infinite symmetric group $S_\infty$ of all permutations of $1, 2, 3,\ldots$ fixing all but a finite number of elements. They form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,\ldots]$ in infinitely many variables.

The cohomology of the flag manifold $\text{Fl}(m)$ is $\mathbb{Z}(x_1,x_2,\ldots,x_m]/I$ , where $I$ is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial $\mathfrak{S}_w$ is the unique homogeneous polynomial of degree $\ell(w)$ representing the Schubert cycle of $w$ in the cohomology of the flag manifold $\text{Fl}(m)$ for all sufficiently large $m$ .

Properties

If $w$ is the permutation of longest length in $S_n$ then $\mathfrak{S}_w = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}^1$
$\partial_i \mathfrak{S}_w = \mathfrak{S}_{ws_i}$ if $w(i) > w(i+1)$ , where $s_i$ is the transposition $(i, i+1)$ and where $\partial_i$ 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.

$\mathfrak{S}_1 = 1$
If $w$ is the transposition $(n,n+1)$ , then $\mathfrak{S}_w = x_1 + \ldots + x_n$ .
If $w(i) < w(i + 1)$ for all $i \neq r$ , then $\mathfrak{S}_w$ is the Schur polynomial $s_\lambda(x_1,\ldots,x_r)$ where $\lambda$ is the partition $(w(r) - r, \cdots, w(2) - 2, w(1) - 1)$ . 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.
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.

Double Schubert polynomials

Double Schubert polynomials $\mathfrak{S}_w(x_1,x_2,\ldots, y_1,y_2,\ldots)$ 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 $\mathfrak{S}_w(x_1,x_2,\ldots, y_1,y_2,\ldots)$ are characterized by the properties

$\mathfrak{S}_w(x_1,x_2,\ldots, y_1,y_2,\ldots) = \prod\limits_{i + j \leq n} (x_i - y_j)$ when $w$ is the permutation on $1,\ldots,n$ of longest length.
$\partial_i \mathfrak{S}_w = \mathfrak{S}_{ws_i}$ 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.

References

Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. (1973), "Schubert cells, and the cohomology of the spaces G/P", Russian Math. Surveys 28: 1–26, doi:10.1070/RM1973v028n03ABEH001557
Fomin, Sergey; Gelfand, Sergei; Postnikov, Alexander (1997), "Quantum Schubert polynomials", Journal of the American Mathematical Society 10 (3): 565–596, doi:10.1090/S0894-0347-97-00237-3, ISSN 0894-0347, MR 1431829
Fulton, William (1992), "Flags, Schubert polynomials, degeneracy loci, and determinantal formulas", Duke Mathematical Journal 65 (3): 381–420, doi:10.1215/S0012-7094-92-06516-1, ISSN 0012-7094, MR 1154177
Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts 35, Cambridge University Press, ISBN 978-0-521-56144-0, MR 1464693
Fulton, William (1999), "Universal Schubert polynomials", Duke Mathematical Journal 96 (3): 575–594, doi:10.1215/S0012-7094-99-09618-7, ISSN 0012-7094, MR 1671215
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