Geometric Nature of Relations on Plabic Graphs and Totally Non-negative Grassmannians

Author:

Abenda Simonetta12,Grinevich Petr G345

Affiliation:

1. Dipartimento di Matematica and Alma Mater Research Center on Applied Mathematics , Università di Bologna, Italy

2. INFN , sez. di Bologna, Italy

3. Steklov Mathematical Institute of Russian Academy of Sciences , Moscow, Russia

4. L.D. Landau Institute for Theoretical Physics , Chernogolovka, Russia

5. Lomonosov Moscow State University , Faculty of Mechanics and Mathematics, Moscow, Russia

Abstract

AbstractThe standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicoloured (plabic) graphs in the disc. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations at the vertices of such graphs, depending on some signatures defined on their edges. The problem of characterizing the signatures corresponding to the totally non-negative cells was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non-negativity property on the full positroid cell. If each edge in a graph $\mathcal G$ belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and the gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of the geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property in the case of bipartite graphs [1, 7], and has a different statistical mechanical interpretation otherwise [6]. An explicit connection between the solution of Lam’s system of relations and the value of Postnikov’s boundary measurement map is established using the generalization of Talaska’s formula [51] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for the transformations of the signatures under Postnikov’s moves and reductions and amalgamations of networks.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3