String cone and superpotential combinatorics for flag and Schubert varieties in type A
String cone and superpotential combinatorics for flag and Schubert varieties in type A
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A 型旗形和舒伯特变种的弦锥和超势组合学
DOI：
10.1016/j.jcta.2019.04.006
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
G. Fourier
中科院分区：
文献类型：
--
作者：
L. Bossinger;G. Fourier
我们研究伪线排列的组合及其与旗帜和舒伯特簇几何的关系。我们将每个伪线排列关联到两个以对偶方式定义的多面体锥体。我们证明了其中之一是 Littelmann 和 Berenstein-Zelevinsky 提出的加权弦锥。对于另一个，我们展示了它是如何在 Gross-Hacking-Keel-Kontsevich 的簇变体和镜像对称的框架中出现的：对于旗形变体，锥体是其超势的热带化，而对于舒伯特变体，超势的限制是必要的。我们证明两个锥体是单模等价的。作为我们组合结果的推论，我们使用聚类理论将舒伯特簇的 Caldero 环面退化视为 GHKK 退化。
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.