Total positivity, Grassmannians, and networks

TL;DR

This paper explores the connection between total positivity, Grassmannians, and planar directed networks, revealing how network boundary measurements parametrize cells in the totally nonnegative Grassmannian and their combinatorial structure.

Contribution

It establishes a natural link between inverse boundary problems for networks and the structure of the totally nonnegative Grassmannian, including cell decomposition and combinatorial descriptions.

Findings

Boundary measurements parametrize Grassmannian cells.

Cell decomposition corresponds to matroid strata.

Descriptions of cell partial order and gluing are provided.

Abstract

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.