Quantum Bruhat graph and tilted Richardson varieties

Quantum Bruhat graph is a weighted directed graph on a finite Weyl group first defined by Brenti-Fomin-Postnikov. It encodes quantum Monk's rule and can be utilized to study the $3$-point Gromov-Witten invariants of the flag variety. In this paper, we provide an explicit formula for the minimal weights between any pair of permutations on the quantum Bruhat graph, and consequently obtain an Ehresmann-like characterization for the tilted Bruhat order. Moreover, for any ordered pair of permutations $u$ and $v$, we define the tilted Richardson variety $T_{u,v}$, with a stratification that gives a geometric meaning to intervals in the tilted Bruhat order. We provide a few equivalent definitions to this new family of varieties that include Richardson varieties, and establish some fundamental geometric properties including their dimensions and closure relations.