Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Total positivity, Grassmannians, and networks

Alexander Postnikov

TL;DR

The paper develops a coherent framework linking total positivity to planar directed networks by realizing the totally nonnegative Grassmannian $Gr_{kn}^{tnn}$ via boundary measurements. It builds a rich array of equivalent combinatorial models—plabic graphs, $\Gamma$-diagrams, decorated permutations, Grassmann necklaces, and alternating strand diagrams—and proves that every nonnegative Grassmann cell has a subtraction-free parametrization arising from a suitable network. The authors establish precise move-equivalences that govern how cells are glued along their boundaries, connect these constructions to cluster theory and string polytopes, and provide comprehensive enumerative results tying together several classical combinatorial objects. The work thus yields a detailed, constructive atlas for the totally nonnegative Grassmannian with broad implications for total positivity, cluster algebras, and related combinatorics.

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.

Total positivity, Grassmannians, and networks

TL;DR

The paper develops a coherent framework linking total positivity to planar directed networks by realizing the totally nonnegative Grassmannian via boundary measurements. It builds a rich array of equivalent combinatorial models—plabic graphs, -diagrams, decorated permutations, Grassmann necklaces, and alternating strand diagrams—and proves that every nonnegative Grassmann cell has a subtraction-free parametrization arising from a suitable network. The authors establish precise move-equivalences that govern how cells are glued along their boundaries, connect these constructions to cluster theory and string polytopes, and provide comprehensive enumerative results tying together several classical combinatorial objects. The work thus yields a detailed, constructive atlas for the totally nonnegative Grassmannian with broad implications for total positivity, cluster algebras, and related combinatorics.

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.

Paper Structure

This paper contains 27 sections, 82 theorems, 69 equations, 41 figures.

Table of Contents

  1. Introduction
  2. Grassmannian
  3. Schubert cells
  4. Plücker coordinates
  5. Matroid strata
  6. Totally nonnegative Grassmannian
  7. Planar networks
  8. Loop-erased walks
  9. $\Gamma$-diagrams
  10. Inverting the boundary measurement map
  11. Lemma on tails
  12. Perfection through trivalency
  13. Forget the orientation
  14. Plabic networks
  15. Transformations of plabic networks
  16. ...and 12 more sections

Key Result

Theorem 3.5

Each totally nonnegative Grassmann cell $S_\mathcal{M}^\mathrm{tnn}$ is homeomorphic to an open ball of appropriate dimension. The decomposition of the totally nonnegative Grassmannian $Gr_{kn}^\mathrm{tnn}$ into the union of the cells $S_\mathcal{M}^\mathrm{tnn}$ is a CW-complex.

Figures (41)

  • Figure 2.1: A Young diagram $\lambda$ and the corresponding subset $I(\lambda)$
  • Figure 4.1: A path $P$ with the winding index $\mathit{wind}(P)=-1$
  • Figure 5.1: Crossings, alignments, and misalignments
  • Figure 6.1: $\Gamma$-property: if $a,c\ne 0$ then $b\ne 0$
  • Figure 6.2: A $\Gamma$-diagram $D$ of shape $\lambda = (5,5,2,1)$ with $|D|=6$
  • ...and 36 more figures

Theorems & Definitions (174)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Example 3.4
  • Theorem 3.5
  • Conjecture 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Lemma 3.9
  • ...and 164 more