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The positive Grassmannian, the amplituhedron, and cluster algebras
Combinatorics · Submitted 21 Oct 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We will explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron.
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The abstract, sentence by sentence
The positive Grassmannian $Gr_{k,n}^{\geq 0}$ is the subset of the real Grassmannian where all Pl\"ucker coordinates are nonnegative. It has a beautiful combinatorial structure as well as connections to statistical physics, integrable systems, and scattering amplitudes. The amplituhedron $\mathcal{A}_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under a positive linear map $\mathbb{R}^n \to \mathbb{R}^{k+m}$. We will explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The positive Grassmannian $Gr_{k,n}^{\geq 0}$ is the subset of the real Grassmannian where all Pl\"ucker coordinates are nonnegative. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s1 | It has a beautiful combinatorial structure as well as connections to statistical physics, integrable systems, and scattering amplitudes. | background 1.00 | 0.07 | 0.06 | asserted 1.00 |
| s2 | The amplituhedron $\mathcal{A}_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under a positive linear map $\mathbb{R}^n \to \mathbb{R}^{k+m}$. | background 0.97 | 0.10 | 0.10 | asserted 1.00 |
| s3 | We will explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron. | contribution 0.82 | 0.50 | 0.25 | asserted 0.93 |
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The paper
Original abstract
The positive Grassmannian $Gr_{k,n}^{\geq 0}$ is the subset of the real Grassmannian where all Pl\"ucker coordinates are nonnegative. It has a beautiful combinatorial structure as well as connections to statistical physics, integrable systems, and scattering amplitudes. The amplituhedron $\mathcal{A}_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under a positive linear map $\mathbb{R}^n \to \mathbb{R}^{k+m}$. We will explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron.
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References
How the literature uses it
10 arXiv papers cite this in 11 sentences
Papers citing it · 4 sentences without a confident label
Cites it as background 7 papers
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A family of Plücker coordinates is said to be weakly separated if all pairs of Plücker coordinates in it are weakly separated [ref] .
Pl\"ucker inequalities for weakly separated coordinates in totally nonnegative Grassmannian · Generalization to pairs of weakly separated Plücker coordinates · 0.94
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The proof of this equality, however, requires introducing the notion of plabic graphs and how they parametrize positroid cells in the Grassmannian $$ [ref] , which exceeds the scope of this paper, and thus will be treated in future work.
Bicolored tilings and the Scott map · Scott map on bicolored tilings · 0.83
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The Amplituhedron has been studied from a purely mathematical perspective as a natural generalization of the positive Grassmannian with many exciting properties [ref] .
Loops of Loops Expansion in the Amplituhedron · Introduction · 0.93
What it cites · 28 sentences without a confident label
Uses the method or tool of 1 paper
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Following [ref] , we now build a cluster variety $_$ in $_k, k+2()$ for each positroid tile $()$ of $_n, k, 2(Z)$.
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers · Positroid tiles as totally positive parts of cluster varieties · 0.84
Compares with 1 paper
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Our notion of positroid subdivision is closely related to the good dissections studied in [ref] .
The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron · $ϕ$-induced subdivisions and positroid tilings · 0.97
Cites as support 2 papers
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However for each $w∈D_k+1,n$ one can find explicit matrices $Z∈_n,k+2^>0$ such that $w$ is nonempty [ref] .
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers · T-duality and positroid tilings of $Δ_k+1,n$ and $𝒜_n,k,2$ · 1.00
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In fact, it is a regular CW decomposition [ref] .
Regularity theorem for totally nonnegative flag varieties · The positroid cell decomposition · 0.99
Cites as background 3 papers
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The totally nonnegative Grassmannian $$ or (informally) the positive Grassmannian [ref] can be defined as the subset of the real Grassmannian $_k,n$ where all Plücker coordinates are nonnegative.
Total positivity, Grassmannians, and networks · Introduction · 0.94
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Acknowledgements: Main references for this article include [ref] .
The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron · Future directions · 0.94
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Acknowledgements: Main references for this article include [ref] .
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers · Future directions · 0.94
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