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We study the combinatorics of pseudoline arrangements and their relation tothe geometry of flag and Schubert varieties. We associate to each pseudolinearrangement two polyhedral cones, defined in a dual manner. We prove that oneof 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 andmirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the coneis the tropicalization of their superpotential while for Schubert varieties arestriction of the superpotential is necessary.  We prove that the two cones are unimodularly equivalent. As a corollary ofour combinatorial result we realize Caldero's toric degenerations of Schubertvarieties as GHKK-degeneration using cluster theory.

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