Positive Grassmannian

Updated 12 June 2026

Positive Grassmannian is the subset of the real Grassmannian where all ordered Plücker coordinates are strictly positive, forming an open cell in the totally nonnegative part.
It is stratified into positroid cells and modeled via combinatorial structures like plabic graphs and Le-diagrams that capture the precise vanishing patterns of Plücker coordinates.
It underpins advancements in cluster algebras, scattering amplitudes, integrable systems, and tropical geometry, linking algebraic and combinatorial phenomena.

The positive Grassmannian, denoted $Gr_{>0}(k, n)$ or $Gr_+(k, n)$ , is the subset of the real Grassmannian parameterizing $k$ -dimensional subspaces of $\mathbb{R}^n$ for which all ordered Plücker coordinates are strictly positive. It forms a rich geometric, combinatorial, and algebraic structure, central to recent advances in cluster algebras, matroid theory, scattering amplitudes, integrable systems, and tropical geometry.

1. Definition and Cell Stratification

The Grassmannian $Gr(k, n)$ is the space of $k$ -planes in $\mathbb{R}^n$ , represented as equivalence classes $[Z]$ of full-rank $k \times n$ matrices modulo left $GL(k)$ -action. For each $Gr_+(k, n)$ 0-subset $Gr_+(k, n)$ 1, the Plücker coordinate is $Gr_+(k, n)$ 2, and these satisfy the quadratic Plücker relations, such as $Gr_+(k, n)$ 3 for $Gr_+(k, n)$ 4.

The positive Grassmannian is

$Gr_+(k, n)$ 5

It is an open cell in the totally nonnegative Grassmannian $Gr_+(k, n)$ 6, where all the Plücker coordinates are non-negative.

Postnikov proved that $Gr_+(k, n)$ 7 is stratified into positroid cells (open positroid varieties) labeled by decorated permutations $Gr_+(k, n)$ 8, satisfying $Gr_+(k, n)$ 9 and $k$ 0. Each cell corresponds to a precise pattern of vanishing and nonvanishing Plücker coordinates compatible with the Plücker relations (Paulos et al., 2014).

2. Combinatorial and Polyhedral Models: Plabic Graphs and Positroid Cells

The positroid stratification is supported by explicit combinatorial models:

Plabic Graphs: These are planar bicolored (black/white) graphs embedded in a disk with $k$ 1 cyclically labeled boundary legs. Each reduced plabic graph (no contractible loops or parallel edges) encodes, via left-right traversal rules, a unique decorated permutation, and thus a positroid cell. Moves such as the square move and merge/unmerge transport between graphs of the same cell label without changing the positroid (Paulos et al., 2014).
Le-diagrams: Equivalently, positroid cells correspond to Le-diagrams—certain fillings of Young diagrams with prescribed zero-one patterns preventing a '01' to the right of a '1' in any row, capturing which Plücker coordinates can be zero (Karpman et al., 2018).
Matroidal Perspective: Each positroid cell is a real positroid variety, the closure of a basis shape locus, determined by a noncrossing transversal matroid presentation. Noncrossingness offers a sufficient and conjecturally necessary criterion for a matroid to be a positroid (Marcott, 2019).

This leads to a rich interplay with polyhedral combinatorics. For example, bridge polytopes $k$ 2 encode the combinatorics of BCFW decompositions of positroid cells and are positive analogs of the classical permutohedron (Williams, 2015). The cell poset reflects both combinatorial and geometric boundary structure, and is closely related to the face-lattice of higher-dimensional associahedra.

3. Cluster Algebra Structure and Plabic-Quiver Duality

The coordinate ring $k$ 3 of the Grassmannian admits a natural cluster algebra structure:

Each reduced plabic graph $k$ 4 for the top cell gives a seed—a quiver $k$ 5 (dual to the faces of $k$ 6) and a set of cluster A-coordinates (Plücker coordinates labeling the faces).
Cluster mutations correspond to local moves (especially the square move) on plabic graphs, yielding exchange relations among cluster variables that reproduce Plücker relations (e.g., $k$ 7).
Lower-dimensional cells correspond to subalgebras, with cluster-equivalence to the coordinate rings of positroid varieties obtained by systematically deleting or freezing nodes in the quiver, tracking vanishing minors (Paulos et al., 2014).

For $k$ 8, the cluster algebra is type $k$ 9, realized combinatorially via triangulations of the $\mathbb{R}^n$ 0-gon; for $\mathbb{R}^n$ 1 it is $\mathbb{R}^n$ 2, with cluster variables and mutations matching the associated generalized associahedron (Paulos et al., 2014).

4. Polyhedral and Tropical Geometry: Hypersimplex, Positroidal Subdivisions, and the Positive Tropical Grassmannian

The positive Grassmannian is closely related to regular positroidal subdivisions of the hypersimplex $\mathbb{R}^n$ 3, defined as $\mathbb{R}^n$ 4. The secondary fan of $\mathbb{R}^n$ 5 coincides with the positive tropical Grassmannian $\mathbb{R}^n$ 6.

Tropical Structure: The $\mathbb{R}^n$ 7 is defined by the positive parts of the tropical Plücker relations: for every $\mathbb{R}^n$ 8-tuple, the minimum among $\mathbb{R}^n$ 9, $Gr(k, n)$ 0, $Gr(k, n)$ 1 is attained at least twice, with the minimal pair drawn from those with positive signs. The positive Dressian is cut out by these relations, and $Gr(k, n)$ 2 (Speyer et al., 2020).
Positroid polytopes: Any positroid polytope is characterized by the property that all its two-dimensional faces are again positroid polytopes (Lukowski et al., 2020).
Polyhedral Subdivisions: Each regular positroidal subdivision of $Gr(k, n)$ 3 arises from a positive tropical Plücker vector and corresponds to a positroid tiling, i.e., a covering of the hypersimplex by positroid polytopes with disjoint interiors (Postnikov, 2018).

This framework also governs the classification of asymptotic soliton solutions to the KP equation, where, for $Gr(k, n)$ 4, soliton graphs correspond exactly to triangulations of the $Gr(k, n)$ 5-gon, and for $Gr(k, n)$ 6, to plabic graphs and maximal weakly separated collections (Karpman et al., 2018).

5. Applications: Scattering Amplitudes, Integrable Systems, and Amplituhedra

The positive Grassmannian is a cornerstone of modern approaches to scattering amplitudes in planar $Gr(k, n)$ 7 super-Yang-Mills theory:

On-shell Diagrams: Each planar on-shell diagram corresponds to a positroid cell; BCFW bridge moves correspond to adjacent transpositions in permutations labeling the cells (Arkani-Hamed et al., 2012). The measure on each cell is naturally expressed as a product of $Gr(k, n)$ 8 forms in positive variables, reflecting Yangian invariance and factorization properties.
Amplituhedron: The amplituhedron $Gr(k, n)$ 9 is the image of $k$ 0 under a totally positive linear map $k$ 1, generalizing cyclic polytopes and encoding all tree-level scattering amplitudes (Williams, 2021). The positroid stratification descends to canonical tilings of the amplituhedron.
KP Solitons: In integrable systems, the leading-order asymptotic behavior of KP soliton solutions is governed by the stratification of $k$ 2, with combinatorial data directly controlling soliton interaction patterns (Karpman et al., 2018).

These structures provide a uniform framework for both combinatorial enumeration of amplitude terms (via positroid tilings or plabic graphs) and analytic calculation (through $k$ 3 measures and positive parameterizations).

6. Minor Arrangements, Weak Separation, and Symmetry Properties

The configuration of equalities and inequalities among Plücker coordinates yields further stratifications on the positive Grassmannian:

Largest Minors: Arrangements of equal largest minors are indexed by sorted sets; they correspond to simplices of the alcoved triangulation of the hypersimplex and are counted by Eulerian numbers.
Smallest Minors: Arrangements of equal smallest minors correspond (for $k$ 4 or $k$ 5) to maximal weakly separated sets, which are precisely the clusters of the associated cluster algebra and realized by face labels of reduced plabic graphs.
Non-weakly Separated Cases: For $k$ 6, plabic graph "chain reactions" can realize additional arrangements of smallest minors beyond weakly separated sets, controlled by Dyck-path interlacing patterns (Farber et al., 2015).
Symmetry and Extremal Problems: The cyclic symmetry of $k$ 7 ensures the regular polygon minimizes the ratio of largest to smallest Plücker coordinate among all points, with uniqueness depending on $k$ 8 (Ogranovich, 2020).

7. Positive Geometries and Current Developments

Cutting the positive Grassmannian by additional linear inequalities yields new positive geometries beyond the classical simplex or amplituhedron. For example, intersections with five or six hyperplanes in $k$ 9 (the positive pentahedron or hexahedron) may or may not admit unique adjoint forms (canonical forms with prescribed residues on facets), and this stratification leads to open questions about classification and explicit construction of such positive geometries (Pavlov et al., 3 Mar 2025).

Furthermore, the connection to tropical and polyhedral models extends to the study of weighted blade arrangements and the identification of extremal rays of the positive tropical Grassmannian, supporting an emerging hierarchy of elementary subdivision types with combinatorial interpretation (Early, 2020).

The positive Grassmannian thus lies at the nexus of geometry, combinatorics, and physics, organizing the algebraic, polyhedral, and tropical structures underlying modern theories of total positivity, cluster algebras, scattering amplitudes, integrable models, and their discrete and continuous limits. Its stratification by positroids, cluster subalgebras, polyhedral subdivisions, and tropical fans provides a unifying framework for the algebraic and combinatorial phenomena observable in nonnegative real algebraic geometry and high-energy theoretical physics (Paulos et al., 2014, Karpman et al., 2018, Farber et al., 2015, Postnikov, 2018, Arkani-Hamed et al., 2012, Williams, 2021, Marcott, 2019, Lukowski et al., 2020, Speyer et al., 2020, Pavlov et al., 3 Mar 2025, Ogranovich, 2020, Early, 2020).