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 $w_{0}$ is the permutation of longest length in $S_{n}$ then ${\mathfrak {S}}_{{w_{0}}}=x_{1}^{{n-1}}x_{2}^{{n-2}}\cdots x_{{n-1}}^{1}$
$\partial _{i}{\mathfrak {S}}_{w}={\mathfrak {S}}_{{ws_{i}}}$ 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 $(P-s_{i}P)/(x_{i}-x_{{i+1}})$ .

Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that ${\mathfrak {S}}_{w}=\partial _{{w^{{-1}}w_{0}}}x_{1}^{{n-1}}x_{2}^{{n-2}}\cdots x_{{n-1}}^{1}$ .

Other properties are

$\mathfrak{S}_{id} = 1$
If $s_{n}$ is the transposition $(n,n+1)$ , then ${\mathfrak {S}}_{{s_{n}}}=x_{1}+\ldots +x_{n}$ .
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.

As an example $\mathfrak{S}_{24531}(x) = x_1 x_3^2 x_4 x_2^2+x_1^2 x_3 x_4 x_2^2+x_1^2 x_3^2 x_4 x_2$ .

Multiplicative structure constants

Since the Schubert polynomials form a basis, there are unique coefficients $c_{{\beta \gamma }}^{{\alpha }}$ such that ${\mathfrak {S}}_{\beta }{\mathfrak {S}}_{\gamma }=\sum _{\alpha }c_{{\beta \gamma }}^{{\alpha }}{\mathfrak {S}}_{\alpha }$ . 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 ${\mathfrak {S}}_{w}(x_{1},x_{2},\ldots ,y_{1},y_{2},\ldots )$ 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 $w(i)>w(i+1)$ .

The double Schubert polynomials can also be defined as $\mathfrak{S}_w(x,y) =\sum_{ w=v^{-1}u \text{ and } \ell(w)=\ell(u)+\ell(v) } \mathfrak{S}_u(x) \mathfrak{S}_v(-y)$ .

Quantum Schubert polynomials

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

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
Lascoux, Alain (1995), "Polynômes de Schubert: une approche historique", Discrete Mathematics, 139 (1): 303–317, doi:10.1016/0012-365X(95)93984-D, ISSN 0012-365X, MR 1336845
Lascoux, Alain; Schützenberger, Marcel-Paul (1982), "Polynômes de Schubert", Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 294 (13): 447–450, ISSN 0249-6291, MR 660739
Lascoux, Alain; Schützenberger, Marcel-Paul (1985), "Schubert polynomials and the Littlewood-Richardson rule", Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, 10 (2): 111–124, doi:10.1007/BF00398147, ISSN 0377-9017, MR 815233
Macdonald, I. G. (1991), "Schubert polynomials", in Keedwell, A. D., Surveys in combinatorics, 1991 (Guildford, 1991), London Math. Soc. Lecture Note Ser., 166, Cambridge University Press, pp. 73–99, ISBN 978-0-521-40766-3, MR 1161461
Macdonald, I.G. (1991b), Notes on Schubert polynomials, Publications du Laboratoire de combinatoire et d'informatique mathématique, 6, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Université du Québec a Montréal, ISBN 978-2-89276-086-6
Manivel, Laurent (2001) [1998], Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, 6, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2154-1, MR 1852463
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