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Quantum Bruhat graph and Schubert polynomials

Author: Alexander Postnikov
Journal: Proc. Amer. Math. Soc. 133 (2005), 699-709
MSC (2000): Primary 05E05, 14N35, 14M15
Published electronically: September 29, 2004
MathSciNet review: 2113918
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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 $G/B$. 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 $A$. 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.

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Additional Information

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

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.

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