Vol. 9, No. 3, 2016

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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

We study some combinatorial objects related to the flag manifold X of Lie type G2. 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.

quantum cohomology, Schubert polynomial, $G_2$ flag manifold
Mathematical Subject Classification 2010
Primary: 14N15
Secondary: 14M15, 14N35, 05E15
Received: 18 February 2015
Accepted: 29 May 2015
Published: 3 June 2016

Communicated by Jim Haglund
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