Geometric nature of relations on plabic graphs and totally nonnegative Grassmannians
Abstract
The standard parametrization of totally nonnegative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicolored (plabic) graphs in the disk. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations on vectors on such graphs, depending on some signatures defined on edges. The problem of characterizing the signatures corresponding to the totally nonnegative 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 nonnegativity property on the full positroid cell. If the graph $\mathcal G$ satisfies the following natural constraint: each edge 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 gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of 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 Kasteleyntype property [7,1] and we conjecture a mechanicalstatistical interpretation of such relations. An explicit connection between the solution of Lam system of relations and the value of Postnikov boundary measurement map is established using the generalization of Talaska formula [50] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtractionfree denominators. Finally, we provide explicit formulas for transformations of signatures under Postnikov moves and reductions, and amalgamations of networks.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2111.05782
 arXiv:
 arXiv:2111.05782
 Bibcode:
 2021arXiv211105782A
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 14M15;
 05C10;
 05C50;
 15B48
 EPrint:
 39 pages, several figures. We have split preprint arXiv:1908.07437 into two parts. This is the fully revised second part. arXiv admin note: text overlap with arXiv:2108.03229