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

2016/11/20 by Bossinger, Lara, Fourier, Ghislain · 2 citations
#13F60 #14M15 #14M25 #14M99 #17B10 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1611.06504

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.

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• Structure and geometry of the tableaux algebra
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