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 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.^{[citation needed]}

Properties

• If w is the permutation of longest length in S_n then 𝔖_w = xn−1
  1xn−2
  2...x1
  n-1
• ∂_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.
• 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 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

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.

See also

• Kostant polynomial
• Monk's formula gives the product of a linear Schubert polynomial and a Schubert polynomial.
• nil-Coxeter algebra

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 
• Sottile, Frank (2001), "s/s130110", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4