---
title: "Total positivity, Grassmannians, and networks"
authors: ["Alexander Postnikov"]
year: 2006
venue: "preprint"
arxiv: "math/0609764"
doi: null
source: "https://arxiv.org/abs/math/0609764"
topics: [curiosity]
seed_rank: 1123
seed_batch: "curiosity-2026-08-21"
reviewed: "2026-08-21"
pool: "maths-foundations"
relevance_score: 9
lineage: total-positivity-positroids
cites:
  - title: "Cluster algebras I: Foundations"
    url: "https://arxiv.org/abs/math/0104151"
    year: 2002
    arxiv: "math/0104151"
    doi: "10.1090/S0894-0347-01-00385-X"
  - title: "The honeycomb model of GL(n) tensor products I: Proof of the saturation conjecture"
    url: "https://arxiv.org/abs/math/9807160"
    year: 1999
    arxiv: "math/9807160"
    doi: "10.1090/S0894-0347-99-00299-4"
  - title: "Positroid cluster structures from relabeled plabic graphs"
    url: "https://arxiv.org/abs/2006.10247"
    year: 2020
    arxiv: "2006.10247"
    doi: null
  - title: "Total positivity: tests and parametrizations"
    url: "https://arxiv.org/abs/math/9812378"
    year: 2001
    arxiv: "math/9812378"
    doi: null
see:
  - "1086-cluster-algebras-i-foundations"
  - "1029-a-rig-of-transformations"
  - "951-an-introduction-to-string-diagrams-for-computer-scientists"
  - "018-physics-topology-logic-and-computation-a-rosetta-stone"
---

# Total positivity, Grassmannians, and networks

## One-sentence takeaway

Points of the totally nonnegative Grassmannian are boundary measurements of planar directed networks (plabic graphs); the cells are positroids, and the measurements are subtraction-free rational functions.

## Why it matters here

Why you might love this: a planar wiring diagram with positive edge weights *is* a point in \(\mathrm{Gr}_{\ge 0}(k,n)\). Switching a wiring (square move, spike removal) is cluster mutation; the subtraction-free formulae are the Laurent phenomenon drawn as a network. Lusztig's total positivity, Fomin–Zelevinsky clusters (1086 already cites this paper), and scattering on-shell diagrams all meet in one picture. Bridge: Anoptic/ano already loves string diagrams and planar rewrites — plabic graphs are the Grassmannian as a planar rewrite system, and a positroid cell is a combinatorial chart on a configuration space.

## Key ideas

- A planar directed network with sources/sinks on a disk has a boundary-measurement matrix whose maximal minors (Plücker coordinates) are subtraction-free rational functions of the edge weights.
- The image is the totally nonnegative Grassmannian \(\mathrm{Gr}_{\ge 0}(k,n)\). Cells are indexed by positroids / decorated permutations / reduced plabic graphs.
- Moves on plabic graphs (square move, merge/split, urban renewal) realise cluster mutation of the coordinate ring; reduced graphs are clusters.
- Unifies Lusztig's dual canonical basis / total positivity with the cluster charts on Grassmannians that cluster algebras I axiomatised.

## Caveats

- 2006 preprint; never journal-published in this form. The theorems are standard and widely used; later cluster-structure proofs (Scott, Leclerc–Zelevinsky, Oh–Postnikov–Speyer, Fraser–Sherman-Bennett) are increments — cite, do not remint.
- Planar networks; non-planar / higher-genus analogues are a different literature.
- Not a scattering-amplitudes paper; the physics reading (Arkani-Hamed–Trnka) is later.

## Links

- arXiv: [math/0609764](https://arxiv.org/abs/math/0609764)