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

Lara Bossinger, Ghislain Fourier

Published 2016-11-20Version 1

We show that every weighted string cone for a reduced expression of the longest element in $S_{n+1}$ is unimodularly equivalent to the cone obtained through the tropicalization of a Landau-Ginzburg superpotential in an appropriate seed of the cluster algebra $\mathbb C[G^{e,w_0}]$, the coordinate ring of the double Bruhat cell $G^{e,w_0}$. We provide, using Gleizer-Postnikov paths in pseudoline arrangements, a combinatorial model to describe the superpotential cones (non-recursively). Our results implies that any toric degeneration of the flag variety obtained through string polytopes, appears also in the framework of potentials on cluster varieties as conjectured by Gross, Hacking, Keel, and Kontsevich. We extend this result also to Schubert varieties and double Bruhat cells $G^{e,w}$, e.g. the weighted string cone of every Schubert variety in type A can be obtained through restricting the superpotential to an appropriate double Bruhat cell.