#### 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
##### 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