论文标题
Kasteleyn定理，磁盘中平面两分网络上的几何特征和KP-II除数
Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk
论文作者
论文摘要
在磁盘计数二分型二聚体配置中具有规定边界条件的平面两分图上的Kasteleyn符号矩阵的最大未成年人，此类矩阵的加权版本提供了对真实草个者完全非负部分的自然参数化（请参阅参考文献。[54,43,444,58,74,58,7]）。在本文中，我们提供了Kasteleyn定理的这种变体的几何解释：签名是Kasteleyn，并且仅在参考的意义上是几何时。 [5]。我们将这种几何表征应用于明确求解相关的关系系统，并提供了一个新的证据，表明由Kasteleyn加权矩阵诱导的阳性细胞的参数化与后尼科夫边界测量图的参数相吻合。最后，我们使用Kasteleyn关系系统将代数几何数据与KP Multi-Soliton解决方案相关联。实际上，如果后者的双重图代表孤子数据，则KP波函数在光谱曲线的节点上求解了这种关系系统。因此，除数的构建自动不变，最后与参考文献相吻合。 [4,6]对于目前的图表。
Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non--negative part of real Grassmannians (see Refs. [54,43,44,58,7]). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Ref. [5]. We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Refs. [4,6] for the present class of graphs.
