Quantum Schubert polynomials for the G2 flag manifold

Elliott, Rachel E.; Lewers, Mark E.; Mihalcea, Leonardo C.

doi:10.2140/involve.2016.9.437

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Quantum Schubert polynomials for the $G_2$ flag manifold

Rachel E. Elliott, Mark E. Lewers and Leonardo C. Mihalcea

Vol. 9 (2016), No. 3, 437–451

DOI: 10.2140/involve.2016.9.437

Abstract

We study some combinatorial objects related to the flag manifold $X$ of Lie type $G_{2}$ . Using the moment graph of $X$ , we calculate all the curve neighborhoods for Schubert classes. We use this calculation to investigate the ordinary and quantum cohomology rings of $X$ . As an application, we obtain positive Schubert polynomials for the cohomology ring of $X$ and we find quantum Schubert polynomials which represent Schubert classes in the quantum cohomology ring of $X$ .

Keywords

quantum cohomology, Schubert polynomial, $G_2$ flag manifold

Mathematical Subject Classification 2010

Primary: 14N15

Secondary: 14M15, 14N35, 05E15

Milestones

Received: 18 February 2015

Accepted: 29 May 2015

Published: 3 June 2016

Communicated by Jim Haglund

Authors

Rachel E. Elliott
Department of Physics Virginia Tech University 306 Robeson Hall Blacksburg, VA 24061 United States

Mark E. Lewers
Department of Mathematics University of Virginia 141 Cabell Drive Kerchof Hall Charlottesville, VA 22904 United States

Leonardo C. Mihalcea
Department of Mathematics Virginia Tech University 460 McBryde Hall Blacksburg, VA 24060 United States

Vol. 9, No. 3, 2016