Global Positroid Varieties

• Global Positroid Varieties are algebraic subvarieties of complex Grassmannians defined using cyclic rank conditions and bounded affine permutations.
• They bridge combinatorial models, total positivity, and representation theory by linking cluster algebra structures, Poisson geometry, and toric degenerations.
• Their study yields practical insights into smoothness criteria, arithmetic point counts, and categorical lifts via birational parametrizations and Johnson graphs.

Global positroid varieties are positroid subvarieties of complex Grassmannians viewed as global algebraic varieties rather than only as cells in the totally nonnegative Grassmannian. In current usage, the term also refers to flat families inside the type $A$ global affine Grassmannian whose general fiber is a classical positroid variety and whose special fiber is a subvariety of the juggling variety (Feigin, 9 Sep 2025). Following Postnikov and Knutson–Lam–Speyer, $\mathrm{Gr}(k,n)$ is stratified into open positroid varieties $\Pi_f^\circ$ indexed by bounded affine permutations $f\in B_{k,n}$, and the closure of each such stratum is a positroid variety (Yost-Wolff, 17 Feb 2026). Equivalently, a positroid variety is the image of a Richardson variety in the full flag variety under the projection to the Grassmannian, and it can also be described as an intersection of cyclically rotated Grassmannian Schubert varieties (Lam, 2018).

1. Classical definitions and combinatorial indexing

A $(k,n)$-bounded affine permutation is a bijection $f:\mathbb{Z}\to\mathbb{Z}$ satisfying

$$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$

The set of all such $f$ is denoted $\operatorname{Bound}(k,n)$ (Lam, 2018). The Grassmannian is stratified into open positroid varieties

$\Pi_f^\circ \subset \mathrm{Gr}(k,n),$

indexed by $\mathrm{Gr}(k,n)$0, and these strata form a disjoint union decomposition of $\mathrm{Gr}(k,n)$1 (Yost-Wolff, 17 Feb 2026).

A second standard parametrization uses Grassmann necklaces. For $\mathrm{Gr}(k,n)$2, the associated necklace $\mathrm{Gr}(k,n)$3 is defined by

$\mathrm{Gr}(k,n)$4

and the map $\mathrm{Gr}(k,n)$5 is a bijection between bounded affine permutations and $\mathrm{Gr}(k,n)$6-Grassmann necklaces (Lam, 2018). In this language the associated positroid is

$\mathrm{Gr}(k,n)$7

where $\mathrm{Gr}(k,n)$8 denotes cyclic rotation (Lam, 2018).

Lam’s representation-theoretic formulation makes the global nature explicit: if $\mathrm{Gr}(k,n)$9, then

$\Pi_f^\circ$0

so a positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties (Lam, 2018). The same family is described by Knutson–Lam–Speyer as the image of a Richardson variety under the projection $\Pi_f^\circ$1 (Ford, 2013). These formulations connect matroid combinatorics, total positivity, and projected Richardson geometry.

A third characterization uses cyclic interval rank conditions. A rank-$\Pi_f^\circ$2 matroid $\Pi_f^\circ$3 on $\Pi_f^\circ$4 is a positroid iff it is generated by rank conditions on cyclic intervals, equivalently iff it is the matroid of a configuration in $\Pi_f^\circ$5 with all $\Pi_f^\circ$6 minors nonnegative, equivalently iff its matroid variety is the image of a Richardson variety in the full flag variety (Ford, 2013). This equivalence is the basic reason positroid varieties occupy a distinguished place among matroid varieties.

2. Codimension, cyclic rank data, and global combinatorics

Ford introduced the expected codimension of a matroid variety as a purely combinatorial invariant. For a rank-$\Pi_f^\circ$7 matroid $\Pi_f^\circ$8 on $\Pi_f^\circ$9, define

$f\in B_{k,n}$0

and recursively define coefficients $f\in B_{k,n}$1 by inclusion–exclusion; then

$f\in B_{k,n}$2

For arbitrary matroids, $f\in B_{k,n}$3 need not equal the actual codimension, as shown by the Pappus matroid, where $f\in B_{k,n}$4 but $f\in B_{k,n}$5 (Ford, 2013).

For positroids, the situation is exact. If $f\in B_{k,n}$6 is a positroid, then

$f\in B_{k,n}$7

and this codimension can be computed from cyclic intervals alone (Ford, 2013). Writing $f\in B_{k,n}$8 for the set of cyclic intervals,

$f\in B_{k,n}$9

Ford proves $(k,n)$0, and $(k,n)$1 is exactly the indicator of a $(k,n)$2 in the affine permutation matrix of the associated bounded affine permutation (Ford, 2013). The resulting sum equals the affine permutation length $(k,n)$3, recovering the known codimension formula

$(k,n)$4

A complementary combinatorial codimension formula uses decorated permutations and chord diagrams. For a decorated permutation $(k,n)$5, the codimension of the associated positroid variety is

$(k,n)$6

This identifies codimension with the number of alignments in the chord diagram and simultaneously with Bruhat interval data (Billey et al., 2022). The same paper proves a Bruhat interval characterization: $(k,n)$7 so the bases of the positroid are exactly the initial $(k,n)$8-sets of permutations in the associated Grassmann interval (Billey et al., 2022).

These formulas place the dimension theory of global positroid varieties entirely inside combinatorics: cyclic intervals, affine permutations, decorated permutations, and Bruhat intervals all recover the same codimension data.

3. Coordinate rings, birational charts, and canonical bases

Global positroid varieties admit several compatible algebraic models. Karpman studies birational parametrizations

$(k,n)$9

for $f:\mathbb{Z}\to\mathbb{Z}$0, and proves that two major constructions coincide: boundary measurement maps for bridge graphs and projected Marsh–Rietsch parametrizations of Deodhar components (Karpman, 2014). In this sense, each positroid variety has a combinatorial atlas of birational torus charts, simultaneously visible from planar networks and from the flag variety.

Galashin and Lam identify the coordinate ring of an open positroid variety with a cluster algebra. For $f:\mathbb{Z}\to\mathbb{Z}$1, the open positroid variety

$f:\mathbb{Z}\to\mathbb{Z}$2

is smooth and affine, and the map sending face variables of a Postnikov diagram to face-labeled Plücker coordinates extends to an isomorphism

$f:\mathbb{Z}\to\mathbb{Z}$3

Thus the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram (Galashin et al., 2019).

For closed positroid varieties, Lam gives a representation-theoretic model of the homogeneous coordinate ring. If $f:\mathbb{Z}\to\mathbb{Z}$4 and $f:\mathbb{Z}\to\mathbb{Z}$5, the cyclic Demazure module is

$f:\mathbb{Z}\to\mathbb{Z}$6

and its crystal is the corresponding intersection of rotated Demazure crystals (Lam, 2018). The degree-$f:\mathbb{Z}\to\mathbb{Z}$7 piece of the homogeneous ideal satisfies

$f:\mathbb{Z}\to\mathbb{Z}$8

and $f:\mathbb{Z}\to\mathbb{Z}$9 has basis given by the surviving dual canonical basis elements indexed by the cyclic Demazure crystal (Lam, 2018). This gives a canonical-basis description of the global coordinate ring compatible with cyclic symmetry and total positivity.

Together, these results show that open positroid varieties are controlled by cluster algebras and closed positroid varieties by cyclic Demazure modules, while bridge-graph and Deodhar parametrizations furnish explicit birational models.

4. Smoothness, tangent spaces, and Poisson geometry

The smoothness problem for positroid varieties has a complete combinatorial answer. For a positroid $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$0 associated with a decorated permutation $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$1, the following are equivalent: $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$2 is smooth; the Johnson graph $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$3 of the positroid is regular; $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$4 has no crossed alignments; the chord diagram $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$5 is a disjoint union of spirographs; and the positroid is a direct sum of uniform matroids (Billey et al., 2022). The same paper computes tangent-space codimension at torus-fixed points using the induced Johnson graph: $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$6 so local singularity detection reduces to adjacency counts in $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$7 (Billey et al., 2022).

A different layer of geometry comes from Poisson structures. The standard Poisson bivector on $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$8 can be identified with a Feigin–Odesskii Poisson structure arising from the moduli stack of complexes on a Kodaira cycle $$f(i+n)=f(i)+n,\qquad \sum_{i=1}^n f(i)=\binom{n+1}{2}+kn,\qquad i\le f(i)\le i+n.$$9, via the shifted Poisson structure on that moduli space (Hua et al., 2024). In this framework, open positroid varieties coincide with the $f$0-leaves of the standard Poisson structure: $f$1 and each $f$2 is a single $f$3-orbit of symplectic leaves (Hua et al., 2024).

For each $f$4, the paper constructs a smooth surjective morphism from $f$5 to an algebraic torus of dimension $f$6, whose fibers are the symplectic leaves of the standard Poisson structure in $f$7 (Hua et al., 2024). The parameter $f$8 is the number of indecomposable summands of the associated bundle on $f$9. This yields a classification of symplectic leaves inside all open positroid varieties and identifies the moduli stack of symplectic leaves of $\operatorname{Bound}(k,n)$0 as an open substack of the stack of vector bundles on $\operatorname{Bound}(k,n)$1 (Hua et al., 2024).

5. Degenerations, fence complexes, and global families

A recent polyhedral approach associates to each positroid variety in $\operatorname{Bound}(k,n)$2 a fence complex, a union of faces of the Gelfand–Tsetlin polytope $\operatorname{Bound}(k,n)$3 (Chang et al., 11 Jun 2026). Fence complexes are homeomorphic to closed balls, they endow $\operatorname{Bound}(k,n)$4 with the structure of a regular CW complex, and this gives a polyhedral complex presentation of the regular CW complex structure on $\operatorname{Bound}(k,n)$5 (Chang et al., 11 Jun 2026). For a positroid variety $\operatorname{Bound}(k,n)$6, the Ehrhart polynomial of the fence complex equals the Hilbert polynomial of $\operatorname{Bound}(k,n)$7, and under the Sturmfels–Gonciulea–Lakshmibai degeneration of $\operatorname{Bound}(k,n)$8, $\operatorname{Bound}(k,n)$9 degenerates to the reduced union of toric varieties corresponding to its fence complex (Chang et al., 11 Jun 2026).

The 2025 paper titled “Global positroid varieties” introduces a different global construction. Positroid varieties admit natural embedding into quiver Grassmannians for equioriented cyclic quivers, and varying the representation produces families

$\Pi_f^\circ \subset \mathrm{Gr}(k,n),$0

inside the type $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$1 global affine Grassmannian (Feigin, 9 Sep 2025). The general fiber $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$2 for $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$3 is isomorphic to the classical positroid variety $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$4, while the special fiber $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$5 is a subvariety of the juggling variety (Feigin, 9 Sep 2025).

These global positroid families are flat, and the special fiber satisfies

$\Pi_f^\circ \subset \mathrm{Gr}(k,n),$6

where $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$7 denotes cyclic rotation of the juggling pattern (Feigin, 9 Sep 2025). The irreducible components of $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$8 are affine Richardson varieties $\Pi_f^\circ \subset \mathrm{Gr}(k,n),$9, indexed by $\mathrm{Gr}(k,n)$00 with $\mathrm{Gr}(k,n)$01, and $\mathrm{Gr}(k,n)$02 (Feigin, 9 Sep 2025). The explicit defining ideal of the family is conjecturally reduced; this is proved completely for $\mathrm{Gr}(k,n)$03 (Feigin, 9 Sep 2025).

This pair of viewpoints—toric degeneration via fence complexes and affine degeneration via global affine Grassmannians—places positroid varieties in two distinct global degeneration theories.

6. Arithmetic, categorification, and higher analogues

The top-dimensional open positroid variety

$\mathrm{Gr}(k,n)$04

with $\mathrm{Gr}(k,n)$05, plays a special role in arithmetic geometry (Yost-Wolff, 17 Feb 2026). When $\mathrm{Gr}(k,n)$06,

$\mathrm{Gr}(k,n)$07

equivalently

$\mathrm{Gr}(k,n)$08

and this is rederived by comparing a split torus quotient with an anisotropic torus quotient (Yost-Wolff, 17 Feb 2026). The main technical input is that cyclic rotation acts trivially on the compactly supported cohomology of the torus quotient $\mathrm{Gr}(k,n)$09 in the coprime case (Yost-Wolff, 17 Feb 2026).

Riordan gives a categorification of cluster structures on lifts of open positroid varieties using Cohen–Macaulay modules. Subcategories $\mathrm{Gr}(k,n)$10 lift Leclerc’s categories $\mathrm{Gr}(k,n)$11, are Frobenius and stably $\mathrm{Gr}(k,n)$12-CY, carry cluster characters, and induce a cluster structure in lifts of open positroid varieties (Riordan, 13 Jun 2026). This places open positroid geometry inside the global $\mathrm{Gr}(k,n)$13-CY categorification of the Grassmannian.

A flag-theoretic extension appears in the theory of flag positroids. For consecutive ranks $\mathrm{Gr}(k,n)$14, the nonnegative tropical flag variety equals the nonnegative flag Dressian,

$\mathrm{Gr}(k,n)$15

and its points give coherent subdivisions of flag positroid polytopes into flag positroid polytopes (Boretsky et al., 2022). In the complete flag case this specializes to Bruhat interval polytopes, while for consecutive partial flags it gives a flag analogue of the positroid polytope picture (Boretsky et al., 2022). This suggests a systematic extension of global positroid geometry from Grassmannians to partial flag varieties.