Quantum Schubert polynomials for the $G_2$ flag manifold

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 $X$.