Twists of Plücker Coordinates as Dimer Partition Functions
Twists of Plücker Coordinates as Dimer Partition Functions
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DOI：
10.1007/s00220-015-2493-7
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发表时间：
2013-09
影响因子：
2.4
通讯作者：
Robert J. Marsh;J. Scott
中科院分区：
文献类型：
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作者：
Robert J. Marsh;J. Scott
格拉斯曼Grk的齐次坐标环具有用称为波斯特尼科夫图的平面图定义的簇结构。与此图相对应的聚类完全由pl<s:1> cker坐标组成。我们在Grk，n上引入了一个与Berenstein-Fomin-Zelevinsky-twist相关的扭转映射，并根据与固定Postnikov图相关的簇变量给出了任意pl<s:1> cker坐标的扭转的显式Laurent展开。展开的形式是对波斯特尼科夫图对偶的二部图的加权版本的（缩放的）二聚体配分函数，由plicker坐标确定的边界条件修改。我们还将扭转映射与最大绿色序列联系起来。
The homogeneous coordinate ring of the Grassmannian Grk,nhas a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Grk,n, related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plücker coordinate. We also relate the twist map to a maximal green sequence.