[Paper Review] Total positivity, Grassmannians, and networks

Alexander Postnikov|ArXiv.org|Sep 27, 2006

Advanced Combinatorial MathematicsMathematics30 references498 citations

TL;DR

This paper establishes a deep connection between planar directed networks, total positivity, and the combinatorics of the Grassmannian. It introduces a parametrization of the totally nonnegative Grassmannian via boundary measurements of networks, proves that these measurements yield a cellular decomposition into nonnegative Grassmann cells, and provides multiple combinatorial models—such as $̳$-diagrams, decorated permutations, and alternating strand diagrams—for these cells, with explicit bijections and enumeration formulas.

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.

Motivation & Objective

• To establish a correspondence between planar directed networks and the totally nonnegative Grassmannian using boundary measurements.
• To characterize the image of the boundary measurement map as the totally nonnegative Grassmannian $Gr_{kn}^{\mathrm{tnn}}$.
• To provide a combinatorial cell decomposition of $Gr_{kn}^{\mathrm{tnn}}$ using matroid strata and nonnegative Grassmann cells.
• To describe transformations of networks that preserve boundary measurements, including gauge invariance and edge direction switches.
• To enumerate nonnegative Grassmann cells and relate them to decorated permutations, $̳$-diagrams, and hyperplane arrangements.

Proposed method

• Define the boundary measurement map $\mathit{Meas}$ from planar networks to the Grassmannian $Gr_{kn}$, where $M_{ij}$ is the sum of products of edge weights over all paths from source $b_i$ to sink $b_j$.
• Introduce the concept of loop-erased walks and winding index to handle networks with directed cycles, ensuring the boundary measurements remain subtraction-free rational functions.
• Use $\Gamma$-diagrams—fillings of Young diagrams with 0s and 1s satisfying the $\Gamma$-property—to parametrize nonnegative Grassmann cells $S_{\mathcal{M}}^{\mathrm{tnn}}$.
• Establish a bijection between $\Gamma$-diagrams and decorated permutations via the Grassmann necklace and circular Bruhat order.
• Apply transformations such as gauge transformations and square moves to relate different networks with the same boundary measurements.
• Construct plabic networks (planar bipartite graphs) and use alternating strand diagrams to model the cell decomposition and study their gluing structure.

Experimental results

Research questions

• RQ1 What is the image of the boundary measurement map from planar directed networks to the Grassmannian?
• RQ2 How can the totally nonnegative Grassmannian be decomposed into cells, and what combinatorial objects parametrize these cells?
• RQ3 Which transformations of networks preserve the boundary measurements, and how can networks be reconstructed from their measurements?
• RQ4 What is the number of nonnegative Grassmann cells in $Gr_{kn}^{\mathrm{tnn}}$, and how is it related to combinatorial invariants like decorated permutations?
• RQ5 Are there multiple combinatorial models (e.g., $\Gamma$-diagrams, rook placements, hyperplane arrangements) that enumerate the same class of objects in $Gr_{kn}^{\mathrm{tnn}}$?

Key findings

• The image of the boundary measurement map $\mathit{Meas}$ is exactly the totally nonnegative Grassmannian $Gr_{kn}^{\mathrm{tnn}}$, and the image of networks with fixed combinatorial structure is a nonnegative Grassmann cell $S_{\mathcal{M}}^{\mathrm{tnn}}$.
• The number of nonnegative Grassmann cells in $Gr_{kn}^{\mathrm{tnn}}$ is equal to the number of $\Gamma$-diagrams of shape $\lambda \subseteq (n-k)^k$, and this number is also equal to the number of decorated permutations with a given anti-exceedance set.
• The total number of decorated permutations of size $n$ is $N_n = \sum_{k=0}^n N_{kn}$, satisfying the recurrence $N_n = n \cdot N_{n-1} + 1$ with $N_0 = 1$, and has exponential generating function $\sum_{n \geq 0} N_n \frac{x^n}{n!} = \frac{e^x}{1-x}$.
• The generating function for the dimension of nonnegative cells is $N_{kn}(q) = \sum_D q^{|D|}$, where the sum is over $\Gamma$-diagrams of shape $\lambda \subseteq (n-k)^k$, and $|D|$ is the number of 1s in the diagram.
• There is a bijection between $\Gamma$-diagrams of triangular shape $\lambda = (n,n-1,\dots,1)$ with no 1s in the corner boxes and permutations in $S_n$, with the number of such diagrams equal to $n!$.
• The number of nonnegative cells in $\Omega_\lambda$ equals the number of teuton placements on $\lambda$, rook placements on a skew shape $\kappa_\lambda$, regions of the hyperplane arrangement $A_{w_\lambda}$, and elements in the Bruhat interval $[e, w_\lambda]$, establishing a deep combinatorial equivalence.