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 $\pi :X\to X^{\prime }$ 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 ${\pi }_{\ast }d$ stable maps sending their marked points within the Schubert varieties $\pi (X_i)$ of $X^{\prime }$. 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 $F{l}_{1,n-1}$. We prove a Chevalley formula in $Q{K}_s(F{l}_{1,n-1})$ via geometrical analysis of the space of stale maps to $F{l}_{1,n-1}$ and of its image via evaluation maps.

PhD thesis

Cited by in corpus (1)

• Positivity of minuscule quantum K-theory