String cone and Superpotential combinatorics for flag and Schubert varieties in type A

TL;DR

This paper explores the combinatorics of pseudoline arrangements and their connection to the geometry of flag and Schubert varieties, establishing a link between string cones, cluster varieties, and mirror symmetry.

Contribution

It introduces two polyhedral cones associated with pseudoline arrangements, proves their unimodular equivalence, and connects them to superpotentials and toric degenerations in type A.

Findings

Proves the weighted string cone is associated with pseudoline arrangements.

Shows the second cone arises from cluster varieties and mirror symmetry.

Realizes Caldero's toric degenerations as GHKK-degenerations using cluster theory.

Abstract

We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero's toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory.