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arxiv: 2606.12815 · v1 · pith:FSYVPKEDnew · submitted 2026-06-11 · 🧮 math.AG · math.CO

Fence Complexes and Toric Degenerations of Positroid Varieties

Cameron Chang , Pranav Enugandla , Josephine Hlavinka This is my paper

Pith reviewed 2026-06-27 06:07 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords positroid varietiesfence complexesGelfand-Tsetlin polytopetoric degenerationGrassmannianDemazure modulesarithmetically GorensteinCW complex
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The pith

Positroid varieties in the Grassmannian degenerate to reduced unions of toric varieties indexed by fence complexes under the Sturmfels-Gonciulea-Lakshmibai degeneration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates to each positroid variety a fence complex consisting of unions of faces of the Gelfand-Tsetlin polytope. It shows these complexes are homeomorphic to closed balls and turn the polytope into a regular CW complex that matches the known cell structure on the nonnegative Grassmannian. The work proves the Ehrhart polynomial of each fence complex equals the Hilbert polynomial of the corresponding positroid variety. It then establishes that, under the standard toric degeneration of the Grassmannian, positroid varieties degenerate to the reduced union of the toric varieties of their fence complexes. This framework yields classifications of arithmetically Gorenstein positroid varieties inside hook Schubert varieties and a recursive character formula for cyclic Demazure modules.

Core claim

Fence complexes are unions of faces of the Gelfand-Tsetlin polytope P_{k,n} associated to positroid varieties in Gr(k,n). These complexes are homeomorphic to closed balls, endow the polytope with a regular CW complex structure, and satisfy an Ehrhart-Hilbert polynomial equality with the positroid variety. Under the Sturmfels-Gonciulea-Lakshmibai degeneration of Gr(k,n) to the toric variety of the Gelfand-Tsetlin polytope, positroid varieties degenerate to the reduced union of toric varieties corresponding to their fence complexes.

What carries the argument

The fence complex, defined as a union of faces of the Gelfand-Tsetlin polytope P_{k,n} associated to a fundamental weight ω_k.

If this is right

  • The Ehrhart polynomial of a fence complex equals the Hilbert polynomial of the associated positroid variety.
  • Positroid varieties contained inside hook Schubert varieties are arithmetically Gorenstein precisely when their fence complexes satisfy a listed combinatorial condition.
  • Cyclic Demazure modules admit a recursive character formula that is equivalent to the formula of Almousa, Gao and Huang.
  • The Gelfand-Tsetlin polytope carries a regular CW complex structure induced by the fence complexes of all positroid varieties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The degeneration supplies a combinatorial recipe for computing Hilbert series and other invariants of positroid varieties directly from polyhedral data.
  • The ball topology on fence complexes may allow topological invariants of positroid varieties to be read off from the polytope without reference to the algebraic variety.
  • The same fence-complex construction could be tested on other toric degenerations or on positroid varieties inside partial flag varieties.

Load-bearing premise

The fence complexes satisfy the homeomorphism to closed balls, the regular CW complex structure on the Gelfand-Tsetlin polytope, the equality between Ehrhart and Hilbert polynomials, and the stated degeneration property.

What would settle it

A positroid variety whose flat limit under the Sturmfels-Gonciulea-Lakshmibai degeneration fails to equal the reduced union of the toric varieties of its fence complex.

Figures

Figures reproduced from arXiv: 2606.12815 by Cameron Chang, Josephine Hlavinka, Pranav Enugandla.

Figure 1
Figure 1. Figure 1: The TNN locus of P 1 C , denoted by P 1 ≥0 , is a 1-simplex. The TNN Schubert cell decomposition of P 1 ≥0 consists of (X∅)≥0, which is a point, and (X(1))≥0, which is a ray and therefore not homeomorphic to any R n . In [32], Postnikov introduces the positroid stratification, which is a refinement of the Schubert stratification. The locally closed strata (Πw u ) ◦ , indexed by Bruhat intervals [u, w] with… view at source ↗
Figure 2
Figure 2. Figure 2: An SDSS F is a fence diagram if and only if every time F contains either of the first two arrangements of segments (possibly with more segments present), it actually contains the last one. We refer to these segment configurations as the/an [L], [ L ], and box, respectively. An SDSS is a fence diagram if and only if it contains no bad configurations. Remark 3.2. Recall a reverse plane partition of shape λ/µ… view at source ↗
Figure 3
Figure 3. Figure 3: A reduced fence diagram F with colF = (s1s2)(s1)() and rowF = (s2s1)(s2)(). Remark 3.6. Given a fence diagram F, let F ⊤ denote the fence diagram whose underlying skew￾shape is the transpose of that of F and whose fences go between the images of the cells of F which are connected by fences. Then, one sees that rowF = w0colF ⊤ w0, and similarly, colF = w0rowF ⊤ w0 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The polytope O(P2,4) is the subset of R 4 x1,x2,x3,x4 consisting of all fillings of either of the above diagrams with xi ’s that satisfy the depicted inequalities. The diagram on the left gives the GT polytope, and fillings of it consisting of integers are called Gelfand–Tsetlin patterns. The diagram on the right gives O(P2,4) as we defined it. The identification between the GT polytope and O(P2,4) proceed… view at source ↗
Figure 5
Figure 5. Figure 5: A graphical depiction of the last step in the proof of Lemma 3.30. Induc￾tion shows that the fence component A contains all fences in both the purple and blue rectangles, so A contains every cell in the big rectangle. By a taxicab path between cells a and b, we mean a collection of fences forming a minimum length path from a to b. Such a path must necessarily have length equal to the taxicab distance d(a, … view at source ↗
Figure 6
Figure 6. Figure 6: A depiction of the situation described in the proof of Lemma 3.31, when a and b are incomparable. Induction ensures the existence of the blue path from a ′ to b contained in A [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The situations in cases (2a) and (2b), respectively, in the proof of Lemma 3.32 Remark 3.33. Given a fence diagram F of shape λ/µ, the poset Pλ/µ induces a poset structure on (Pλ/µ)/ ∼F , where ∼F is the equivalence relation of belonging to the same fence component. The compatibility from Lemma 3.32 ensures that this is well defined. Then, one readily sees that standard fence tableaux correspond to total e… view at source ↗
Figure 8
Figure 8. Figure 8: The fence diagrams corresponding to the faces of O(P2,4) contained in F 3412 2143 . The fences correspond to the equality conditions cutting out the faces. 4.2. The Hilbert Polynomial of Πw u . We can now connect the Ehrhart function of the fence complex to the Hilbert function of the positroid variety. First, recall that if ∆ is a simplicial complex with vertex set V (∆), then the Stanley–Reisner ring SR(… view at source ↗
Figure 9
Figure 9. Figure 9: Local transformation to go from G (top) to G′ (bottom). Fences of G and G′ are drawn in black, whereas fences of F are drawn in gray. Red dashed edges indicate absence of fences in G and G′ . If there is no edge, it does not matter to the argument whether a fence is there or not. That m1 ≤ j +r −1 is slightly more involved. If row (i) G contains sj+r, find the smallest t ≤ r such that row (i) G contains sj… view at source ↗
Figure 10
Figure 10. Figure 10: Example of a track, with horizontal start and horizontal end. Fences in black belong to both F and G, whereas fences in gray belong only to G. P P Q Q [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: Local transformations converting F to F ′ . The left column shows a track with vertical start and horizontal end, and the right column shows a track with horizontal start and vertical end. Clearly F ′ is a face of F w u containing G, so it suffices to show that wordF′ ≡ wordF . Note applying the above transformation only changes the subwords colT and rowT of wordF , say to col′ T and row′ T . colT is comm… view at source ↗
Figure 14
Figure 14. Figure 14: A generalized antidiagonal, with terms boxed in purple, of a semistan￾dard Young tableau [PITH_FULL_IMAGE:figures/full_fig_p047_14.png] view at source ↗
read the original abstract

We associate to each positroid variety in the Grassmannian $\mathrm{Gr}(k,n)$ a polyhedral complex, which we call a fence complex. Fence complexes consist of unions of faces of the Gelfand-Tsetlin polytope $P_{k,n}$ associated to a fundamental weight $\omega_k$. We show that these fence complexes are homeomorphic to closed balls. Furthermore, they endow the Gelfand-Tsetlin polytope with the structure of a regular CW complex, giving a polyhedral complex presentation of the regular CW complex structure on $\mathrm{Gr}(k,n)_{\geq 0}$. We also show that the Ehrhart polynomial of a fence complex equals the Hilbert polynomial of the associated positroid variety. We prove that under the Sturmfels-Gonciulea-Lakshmibai degeneration of $\mathrm{Gr}(k,n)$ to the toric variety of the Gelfand-Tsetlin polytope, positroid varieties degenerate to the reduced union of toric varieties corresponding to their fence complexes. As an application, we classify when positroid varieties contained inside hook Schubert varieties are arithmetically Gorenstein. We also derive a recursive character formula for cyclic Demazure modules, which we show is equivalent to a formula of Almousa, Gao and Huang.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper associates to each positroid variety in Gr(k,n) a fence complex, defined as a union of faces of the Gelfand-Tsetlin polytope P_{k,n} for the fundamental weight ω_k. It proves that fence complexes are homeomorphic to closed balls, endow P_{k,n} with the structure of a regular CW complex that presents the CW structure on the non-negative Grassmannian, that the Ehrhart polynomial of each fence complex equals the Hilbert polynomial of the corresponding positroid variety, and that under the Sturmfels-Gonciulea-Lakshmibai degeneration positroid varieties degenerate to the reduced union of the toric varieties of their fence complexes. Applications include a classification of arithmetically Gorenstein positroid varieties inside hook Schubert varieties and a recursive character formula for cyclic Demazure modules shown equivalent to a formula of Almousa-Gao-Huang.

Significance. If the stated theorems hold, the work supplies a polyhedral model that realizes positroid varieties as reduced unions of toric varieties under a known degeneration, while also furnishing a CW-complex presentation of the non-negative Grassmannian and matching Ehrhart-Hilbert data. The Gorenstein classification and the Demazure-module character formula are concrete applications that connect the construction to existing results in the literature.

minor comments (2)
  1. [Abstract] The abstract packs several distinct theorems into a single paragraph; separating the statements of the homeomorphism, CW-structure, Ehrhart-Hilbert equality, and degeneration results would improve readability.
  2. Notation for positroid varieties, fence complexes, and the precise faces of P_{k,n} that constitute each complex should be introduced with a short table or diagram in the first section to aid readers unfamiliar with positroid combinatorics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. The referee's assessment correctly identifies the main results on fence complexes, their topological and combinatorial properties, the toric degeneration, and the applications to Gorenstein positroid varieties and Demazure modules.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines fence complexes as new polyhedral objects (unions of faces of the Gelfand-Tsetlin polytope associated to positroids) and proves their topological, combinatorial, Ehrhart, and degeneration properties via direct arguments that build on the external Sturmfels-Gonciulea-Lakshmibai degeneration and standard facts about positroid varieties and Gelfand-Tsetlin polytopes. No step reduces by definition or by self-citation to its own input; the central claims (homeomorphism to balls, CW structure, Ehrhart-Hilbert equality, toric degeneration) are established by theorems whose hypotheses are independent of the conclusions. The equivalence to the Almousa-Gao-Huang formula is shown by explicit recursion rather than by renaming or self-referential citation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

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