Year: 2023
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gao, Jiyang, Gao, Shiliang, Gao, Yibo
Institution: Harvard University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4386501576