Quantum Bruhat graph and Schubert polynomials

Author:
Alexander Postnikov

Journal:
Proc. Amer. Math. Soc. **133** (2005), 699-709

MSC (2000):
Primary 05E05, 14N35, 14M15

DOI:
https://doi.org/10.1090/S0002-9939-04-07614-2

Published electronically:
September 29, 2004

MathSciNet review:
2113918

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Abstract | References | Similar Articles | Additional Information

Abstract: The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of . We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph.

We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quantum Bruhat graph of type . The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes.

**[BFP]**FRANCESCO BRENTI, SERGEY FOMIN, ALEXANDER POSTNIKOV: Mixed Bruhat operators and Yang-Baxter equations for Weyl groups,*International Mathematics Research Notices*, no. 8, 419-441. MR**1999****2000e:20067****[C-F]**IONU¸T CIOCAN-FONTANINE: On quantum cohomology rings of partial flag varieties,*Duke Mathematical Journal*(1999), no. 3, 485-524. MR**98****2000d:14058****[FGP]**SERGEY FOMIN, SERGEI GELFAND, ALEXANDER POSTNIKOV: Quantum Schubert polynomials,*Journal of the American Mathematical Society*(1997), no. 3, 565-596. MR**10****98d:14063****[F-K]**SERGEY FOMIN, ANATOL N. KIRILLOV: Quadratic algebras, Dunkl elements, and Schubert calculus, in ``Advances in Geometry,''*Progress in Mathematics*(1999), 147-182. MR**172****2001a:05152****[F-W]**WILLIAM FULTON, CHRISTOPHER WOODWARD: On the quantum product of Schubert classes, e-print ArXiv`math.AG/0112183`.**[Hum]**JAMES E. HUMPHREYS:*Introduction to Lie Algebras and Representation Theory*, second printing (revised), Graduate Texts in Mathematics**9**, Springer-Verlag, New York, Berlin, 1978. MR**81b:17007****[K-M]**ANATOL N. KIRILLOV, TOSHIAKI MAENO: Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula,*Discrete Mathematics*(2000), no. 1-3, 191-223. MR**217****2001f:05161****[LSc]**ALAIN LASCOUX, MARCEL-PAUL SCHÜTZENBERGER: Polynômes de Schubert,*Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique*(1982), no. 13, 447-450. MR**294****83e:14039****[L-S]**CRISTIAN LENART, FRANK SOTTILE: Skew Schubert polynomials,*Proceedings of the American Mathematical Society*(2003), no. 11, 3319-3328.**131****[Mac]**I. G. MACDONALD:*Schubert polynomials*, Cambridge University Press, Cambridge, 1991. MR**93d:05159****[Pet]**DALE PETERSON:*Lectures on quantum cohomology of*, M.I.T., 1996.**[Po1]**ALEXANDER POSTNIKOV: On a quantum version of Pieri's formula, in ``Advances in Geometry,''*Progress in Mathematics*(1999), 371-383. MR**172****99m:14096****[Po2]**ALEXANDER POSTNIKOV: Symmetries of Gromov-Witten invariants, in ``Advances in Algebraic Geometry Motivated by Physics,''*Contemporary Mathematics*(2001), 251-258. MR**276****2003c:14062****[Po3]**ALEXANDER POSTNIKOV: Affine approach to quantum Schubert calculus, e-print ArXiv`math.CO/0205165`.

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Additional Information

**Alexander Postnikov**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
apost@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07614-2

Keywords:
Quantum cohomology,
flag manifold,
Schubert polynomials.

Received by editor(s):
July 21, 2003

Received by editor(s) in revised form:
November 20, 2003

Published electronically:
September 29, 2004

Additional Notes:
The author was supported in part by NSF grant DMS-0201494.

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.