{"title":"Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk","authors":"Simonetta Abenda","doi":"10.1007/s11040-021-09405-2","DOIUrl":null,"url":null,"abstract":"<div><p>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 (Postnikov et al. J. Algebr. Combin. <b>30</b>(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) <b>92</b>(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). 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 Abenda and Grinevich (2019). 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 Abenda and Grinevich (Sel. Math. New Ser. <b>25</b>(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09405-2.pdf","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09405-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 6
Abstract
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 (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). 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 Abenda and Grinevich (2019). 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 Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.
具有规定边界条件的盘数二聚体构型中平面二部图上Kasteleyn符号矩阵的极大次元,以及这种矩阵的加权形式提供了实Grassmannians (Postnikov et al.)的完全非负部分的自然参数化。j . Algebr。中华医学杂志,30(2),173-191,2009;Lam J. Lond。数学。Soc。(2) 92(3), 633-656, 2015;林2016;尔2016;Affolter et al. 2019)。在本文中,我们对Kasteleyn定理的这种变体提供了几何解释:一个签名当且仅当它在Abenda和Grinevich(2019)的意义上是几何的,就是Kasteleyn。我们将这一几何表征应用于显式求解关联系统,并提供了新的证据,证明Kasteleyn加权矩阵诱导的正极细胞的参数化与Postnikov边界测量图的参数化一致。最后利用Kasteleyn关系系统将代数几何数据与KP多孤子解关联起来。如果谱曲线的对偶图表示孤子数据,则KP波函数确实在谱曲线的节点处解决了这种关系系统。因此,除数的构造是自动不变的,最后与Abenda和Grinevich (Sel)的构造一致。数学。新学报,25(3),43,2019;Abenda and Grinevich, 2020)对于当前这类图。
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