Totally Nonnegative Grassmannian

• The totally nonnegative Grassmannian is a semialgebraic subset of the real Grassmannian where all Plücker coordinates are nonnegative, ensuring uniform sign conditions after projective normalization.
• It is homeomorphic to a closed ball and exhibits a regular CW complex structure through its stratification into positroid cells, providing clear topological insights.
• Its applications span integrable systems, network models, and combinatorial theory, connecting topics such as KP solitons, plabic graphs, and cluster algebras.

The totally nonnegative Grassmannian, usually denoted $\Gr_{\ge 0}(k,n)$ or $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$, is the semialgebraic subset of the real Grassmannian consisting of $k$-dimensional subspaces whose Plücker coordinates are nonnegative. Equivalently, after projective rescaling, all nonzero Plücker coordinates have the same sign. In Lusztig’s framework it is the Grassmannian specialization of the totally nonnegative part of a partial flag variety $G/P$, and it is a central object in combinatorics, representation theory, and mathematical physics (Galashin et al., 2018, Karp, 2015).

1. Foundational definitions

For the real Grassmannian $\Gr(k,n)$, a point $X$ may be represented by a full-rank $k\times n$ matrix, and its Plücker coordinates $\Delta_I(X)$, indexed by $I\in\binom{[n]}{k}$, are the maximal minors of such a representative. The totally nonnegative Grassmannian is

$\Gr_{\ge 0}(k,n)=\{X\in \Gr(k,n): \Delta_I(X)\ge 0 \text{ for all } I\in \tbinom{[n]}{k}\},$

while the totally positive part $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$0 is defined by strict positivity of all Plücker coordinates. The condition is independent of the chosen representative: for some, and thus any, representative matrix, all Plücker coordinates are nonnegative (Galashin et al., 2017).

Lusztig’s general construction begins with a split semisimple algebraic group $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$1, its flag variety $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$2, and a partial flag variety $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$3. The totally nonnegative part of $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$4 is defined as

$\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$5

where $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$6 is the natural projection and $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$7 is the closure of the totally positive part $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$8. For $\mathrm{Gr}_{k,n}^{\mathrm{tnn}}$9 and $k$0 the stabilizer of a $k$1-plane, one recovers the Grassmannian case $k$2 and the Plücker-coordinate description above (Galashin et al., 2018).

This Plücker-coordinate viewpoint coexists with several equivalent formulations. One classical characterization states that a subspace $k$3 is totally nonnegative precisely when all nonzero Plücker coordinates have the same sign; by choosing a projective normalization, one may take them all to be nonnegative (Karp, 2015).

2. Global topology: the ball theorem

A fundamental topological result is that

$k$4

Galashin, Karp, and Lam proved this first for the Grassmannian by a flow argument, and then in full generality for the totally nonnegative part of any partial flag variety $k$5, showing that $k$6 is homeomorphic to a closed ball of dimension equal to that of $k$7 (Galashin et al., 2017, Galashin et al., 2018).

The proof for $k$8 does not use cell decompositions. Instead, it constructs a contractive flow on a global coordinate chart. In one formulation, the relevant operator is the self-adjoint cyclic shift

$k$9

and the flow is $G/P$0. In matrix coordinates $G/P$1, the action takes the form

$G/P$2

where the exponents are negative, so every nonzero point is strictly contracted toward the distinguished totally positive point $G/P$3 (Galashin et al., 2017).

In the general $G/P$4 setting, the vector field

$G/P$5

induces a contractive flow on $G/P$6. After embedding into $G/P$7 using Lusztig’s canonical basis, the flow acts by exponential rescaling of coordinates,

$G/P$8

with all exponents negative. A key topological lemma states that if a bounded invariant region admits such a contractive flow, then its closure is homeomorphic to a closed ball and its boundary to a sphere (Galashin et al., 2018).

For the Grassmannian, the theorem identifies the closure of the top positroid cell with the entire totally nonnegative Grassmannian. In small cases this topology can be visually polyhedral: for $G/P$9, the resulting ball is an octahedron (Galashin et al., 2018).

3. Positroid stratification and regular CW structure

Postnikov’s decomposition of $\Gr(k,n)$0 into positroid cells is one of the central combinatorial structures attached to the space. The decomposition may be written as

$\Gr(k,n)$1

where $\Gr(k,n)$2 ranges over bounded affine permutations; equivalently, cells may be indexed by positroids, Grassmann necklaces, or related combinatorial models (Lam, 2015).

Each positroid cell is isomorphic to a Euclidean space $\Gr(k,n)$3 for some $\Gr(k,n)$4, and its points share the same positroid, namely the collection of $\Gr(k,n)$5-subsets indexing nonvanishing Plücker coordinates. For the top cell $\Gr(k,n)$6, all Plücker coordinates are strictly positive, and the cell is homeomorphic to $\Gr(k,n)$7 (Feigin et al., 2021, Lam, 2015).

The closure order is combinatorial. In the bounded affine permutation model it is governed by Bruhat order, while in the Grassmann necklace model the map from positroids to necklaces is order-reversing. Feigin, Lanini, and Pütz constructed quiver Grassmannians $\Gr(k,n)$8 whose cells are naturally labeled by Grassmann necklaces and whose cell posets coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannian (Feigin et al., 2021).

A stronger topological theorem refines the global ball theorem: the positroid cell decomposition forms a regular CW complex, and the closure of each positroid cell is homeomorphic to a closed ball. This confirms Postnikov’s conjecture for $\Gr(k,n)$9 and Williams’s conjecture for totally nonnegative flag varieties (Galashin et al., 2019). Thus the entire space is a ball, and every individual cell closure is also a ball, with face relations realized geometrically by the positroid closure poset. While the cells are generally not polyhedral in the usual sense, they are contractible and assemble into a regular CW complex whose topology is as controlled as that of a convex polytope (Galashin et al., 2018).

4. Sign variation, stability, and inequalities

A classical theorem of Gantmakher–Krein and Schoenberg–Whitney gives a non-Plücker characterization of total nonnegativity. For $X$0, let $X$1 be the number of sign changes in a vector $X$2 after ignoring zeros, and let $X$3 be the maximal possible sign variation obtained by assigning nonzero signs to zero entries. Then

$X$4

and

$X$5

This criterion extends to the full Grassmannian through sign-variation bounds on certain sequences of Plücker coordinates, and in the non-generic case there is an explicit perturbation algorithm by adjacent-column operations that produces a generic subspace with the same sign-variation bound (Karp, 2015).

The same paper gives two methods for recovering the positroid cell of a totally nonnegative point from sign patterns of vectors in the subspace. One criterion tests realization of alternating sign vectors on $X$6-subsets, and another identifies the Schubert-cell label as a Gale minimum computed from strict alternation data (Karp, 2015).

A distinct equivalence links total nonnegativity to polynomial stability. If

$X$7

is a homogeneous multiaffine polynomial whose coefficients are the Plücker coordinates of a point $X$8, then $X$9 is stable with respect to the upper half-plane if and only if $k\times n$0. Moreover, a matrix $k\times n$1 preserves stability of multiaffine polynomials under the induced action $k\times n$2 if and only if $k\times n$3 is totally nonnegative (Purbhoo, 2016).

Recent work has sharpened the theory of inequalities on $k\times n$4. For weakly separated pairs of Plücker coordinates, partial sums of the long Plücker relations oscillate around $k\times n$5 yet remain nonnegative on the totally nonnegative Grassmannian; this property characterizes weak separability and connects weakly separated sets, cluster combinatorics, Temperley–Lieb immanants, and additive determinantal inequalities (Soskin et al., 2023). Chevalley operations on index sets subsequently gave a classification of all quadratic inequalities in Plücker coordinates over $k\times n$6, and were used to reprove Lam’s log-supermodularity of Plücker coordinates, with consequences including that each positroid is a distributive lattice (Vishwakarma, 25 Feb 2025).

The sign-variation viewpoint also controls analytic behavior of Wronskians. If $k\times n$7 are linearly independent polynomials of degree at most $k\times n$8 whose coefficient matrix has all nonnegative $k\times n$9 minors, then the Wronskian $\Delta_I(X)$0 has no complex zeros in the sector $\Delta_I(X)$1, and the bound $\Delta_I(X)$2 is tight (Karp, 25 Aug 2025).

5. Network models, plabic graphs, and geometric parametrizations

One of Postnikov’s foundational contributions is the boundary measurement parametrization of $\Delta_I(X)$3 by weighted planar bicolored graphs. In this description, positroid cells admit explicit combinatorial parametrizations via Le-diagrams, network parametrizations, and planar graphs; the coordinates on each cell are positive real numbers, except for coordinates forced to vanish by the cell stratification (Galashin et al., 2018). Lam’s survey emphasizes that every point of the totally nonnegative Grassmannian arises as the boundary measurement of a planar bipartite network with positive edge weights (Lam, 2015).

Abenda and Grinevich studied Lam’s alternative parametrization by linear relations on edge vectors of plabic graphs. They constructed explicit geometric signatures on edges, determined by local winding numbers and intersection numbers relative to a perfect orientation and a gauge ray direction. For graphs in which every edge belongs to an oriented path from boundary to boundary, the resulting signature is unique up to vertex gauge transformation. They also established a Kasteleyn-type property: the total signature around a finite face depends only on the number of white vertices on that face. The edge-vector formulas are rational in the edge weights with subtraction-free denominators, and their construction matches Postnikov’s boundary measurement map through a generalized Talaska formula (Abenda et al., 2021).

These network descriptions have direct consequences in integrable systems. Multiline KP-II soliton solutions are parametrized by points of $\Delta_I(X)$4, and Le-networks encode the associated algebraic-geometric data. For every positroid cell, one obtains a reducible rational $\Delta_I(X)$5-curve and a real regular KP divisor directly from the network, with the genus of the underlying smooth $\Delta_I(X)$6-curve equal to the dimension of the positroid cell. This extends earlier constructions from the top cell to all of $\Delta_I(X)$7 and makes the spectral data canonical in terms of the cell parametrization (Abenda et al., 2018).

6. Related spaces, analogues, and extensions

The totally nonnegative Grassmannian serves as a source for several other positive geometries. The same contractive-flow technique that proves $\Delta_I(X)$8 is a ball also shows that the compactification of the space of planar electrical networks $\Delta_I(X)$9 and the cyclically symmetric amplituhedron are homeomorphic to closed balls of dimensions $I\in\binom{[n]}{k}$0 and $I\in\binom{[n]}{k}$1, respectively (Galashin et al., 2017). More generally, a Grasstope is the image of $I\in\binom{[n]}{k}$2 under a linear map $I\in\binom{[n]}{k}$3; in the case $I\in\binom{[n]}{k}$4, such images can be characterized as unions of cells of a hyperplane arrangement satisfying a sign-variation condition, extending work on amplituhedra (Mandelshtam et al., 2023).

The totally nonnegative Grassmannian also sits inside compactifications and degenerations. The nonnegative configuration space, defined as the totally nonnegative part of the Chow quotient of the Grassmannian, has a stratification by positive Chow cells and is homeomorphic to a polytope as a stratified space. Its strata are canonically indexed by regular subdivisions of the hypersimplex into positroid polytopes, equivalently by cones in the positive tropical Grassmannian and the positive Dressian (Arkani-Hamed et al., 2020). A different boundary geometry appears in totally nonnegative critical varieties $I\in\binom{[n]}{k}$5, which arise as images of affine poset cyclohedra; for the top-dimensional positroid cell, $I\in\binom{[n]}{k}$6 is homeomorphic to the second hypersimplex $I\in\binom{[n]}{k}$7 (Galashin, 2021).

There are also noncommutative and finite-field analogues. In the quantum Grassmannian, “quantum positroids” define torus-invariant completely prime ideals, and the poset of torus-invariant prime ideals mirrors the closure poset of positroid cells in the totally nonnegative Grassmannian (Launois et al., 2019). Over a finite field $I\in\binom{[n]}{k}$8, one may define a finite totally nonnegative Grassmannian by declaring an element nonnegative if it is a square; explicit point counts are available in several cases, but the comparison with the real theory reveals genuine differences, including the failure of the sign-variation criterion and the fact that positivity is not compatible with addition (Machacek, 2024).

Across these developments, the totally nonnegative Grassmannian remains the basic model: a closed ball stratified by positroid cells, governed simultaneously by Plücker nonnegativity, sign-variation bounds, network parametrizations, and a rich hierarchy of algebraic and combinatorial structures.