Curve neighborhoods and minimal degrees in quantum products

Abstract

Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be an arbitrary parabolic subgroup of G. Let be the G-homogeneous projective space attached to this situation. We consider the (small) quantum cohomology ring attached to X. Postnikov proved that any quantum product of two Schubert classes admits a unique minimal degree. In particular, this result implies that there exists a unique degree d which is minimal with the property that qd occurs with nonzero coefficient in the quantum product of two point classes. This unique minimal degree in is denoted by dX . We use the technique of curve neighborhoods by Buch-Mihalcea to compute dX explicitly in terms of Kostant’s cascade of orthogonal roots. Moreover, we prove orthogonality relations in greedy decompositions of minimal degrees. Such orthogonality relations can be seen as a generalization of the computation of dX from one minimal degree to every minimal degree, and were successfully applied to prove quasi-homogeneity of the moduli space of stable maps.