Quantum K theory for flag varieties

Abstract

Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $\pi : X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the variety associated with degree d stable maps sending their marked points within Schubert varieties $X_i$ of $X$ is a rationally connected fibration over its image, which parametrizes degree $\pi_* d$ stable maps sending their marked points within the Schubert varieties $\pi(X_i)$ of $X'$. The Euler characteristic of these varieties are quantum $K$-invariants. Our result implies equalities between quantum $K$ correlators. We extend these equalities to the equivariant setting. Finally, we study the small quantum $K$-ring of the universal hyperplane $Fl_{1,n-1}$. We prove a Chevalley formula in $QK_s(Fl_{1,n-1})$ via geometrical analysis of the space of stale maps to $Fl_{1,n-1}$ and of its image via evaluation maps.