Quantum Bruhat graphs and tilted Richardson varieties

TL;DR

This paper provides explicit formulas for minimal weights in the quantum Bruhat graph, characterizes the tilted Bruhat order, and introduces tilted Richardson varieties with geometric properties and stratifications.

Contribution

It introduces the tilted Richardson varieties and offers explicit formulas and geometric insights into the quantum Bruhat graph and related orders.

Findings

Explicit minimal weight formulas for quantum Bruhat graph

Ehresmann-like characterization of tilted Bruhat order

Fundamental geometric properties of tilted Richardson varieties

Abstract

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.