Séminaire Lotharingien de Combinatoire, 93B.36 (2025), 12 pp.
Jiyang Gao, Shiliang Gao and Yibo Gao
Quantum Bruhat Graphs and Tilted Richardson Varieties
Abstract.
The 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 a combinatorial 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. 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 prove some fundamental geometric properties of this new family of varieties, including their dimensions, closure relations, irreducibility, and a Deodhar-like decomposition. We demonstrate their equivalence to the two-point curve neighborhoods of Schubert varieties
Xu and Xv in the minimal degree, and relate their cohomology classes to quantum products of Schubert classes.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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