The positive orthogonal Grassmannian

Yassine El Maazouz; Yelena Mandelshtam

REVIEW 4 major objections 5 minor 1 cited by

The positive orthogonal Grassmannian

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Pith's one-line read Positroid cells of Gr_+(k,n) do not induce a CW cell decomposition of OGr_+(k,n) for n>2k+1; a triangle cell glues to the diagonal of a square cell.

desk verdict Sections 3-4 are genuinely useful, but the Section 5 obstruction for OGr+(2,6) is not valid as stated, so the advertised negative result is unproven. read the letter →

arxiv 2412.14091 v2 pith:U2QZGCKX submitted 2024-12-18 math.CO math.AG

Yassine El Maazouz , Yelena Mandelshtam This is my paper · ORCID

classification math.COmath.AG MSC 14M1515B4805E14

keywords OrthogonalGrassmannianPositivegeometryPositroidcellsCWcelldecompositionMatchingsGröbnerbasisStraighteninglawsPlückercoordinates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to map out the positive orthogonal Grassmannian $\mathrm{OGr}_+(k,n)$---the isotropic $k$-planes in $\mathbb{C}^n$ with all Plücker coordinates nonnegative---for general $k,n$, beyond the well-studied $n=2k$ case. It proves that for $n=2k+1$ the space is linearly isomorphic to $\mathrm{OGr}_+(k+1,2k+2)$, so its boundary combinatorics are governed by matchings on $[2k+2]$. It also proves $\mathrm{OGr}_+(1,n)$ is a positive geometry combinatorially equivalent to a product of two simplices. The paper's main negative claim is that for $n>2k+1$ and $k>1$, the positroid cells of $\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_+(k,n)$, because in $\mathrm{OGr}_+(2,6)$ a triangular cell's edge is glued to the diagonal of a square cell. If this claim holds, the standard positroid combinatorics must be replaced by a new framework, and the paper introduces orthopositroids as the candidate language.

What carries the argument

The mechanism behind the positive results and the obstruction is the sign-alternating quadratic form $\omega_0(x,y)=x_1y_1-x_2y_2+\cdots+(-1)^{n-1}x_ny_n$ and the equations it imposes on Plücker coordinates, $P\Omega P^T=0$ in the notation of Remark 2.3. These equations force a compatibility condition on positroids: a positroid $M$ can meet $\mathrm{OGr}_+(k,n)$ only if, for every pair of $(k-1)$-element index sets $I,J$, the sets $A^+_{IJ}(M)$ and $A^-_{IJ}(M)$ are either both nonempty or both empty; such $M$ are called orthopositroids. The paper's counterexample is carried by two explicit matrices in $\mathrm{OGr}_+(2,6)$, one giving a triangular cell and one giving a square cell whose closure is isomorphic to $\mathrm{OGr}_+(1,4)$, with the triangle edge glued to the square diagonal. A displayed $(k-2)\times(n-6)$ block extension is then used to transport this glued configuration to all $n>2k+1$ and $k>1$.

What would settle it

Compute $P\Omega P^T$ for the $k\times n$ block matrix displayed in Section 5 with a generic positive $(k-2)\times(n-6)$ block: if the isotropy equations (5) force some Plücker coordinate to change sign or force the block to vanish, then the extension to all $n>2k+1$ and $k>1$ fails, leaving only the $(2,6)$ obstruction. A positive result would confirm the intended embedding and support the non-CW claim in the general regime.

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Extended reading notes

Core claim

The central discovery is that the positroid stratification of the positive Grassmannian does not descend to a CW decomposition of the positive orthogonal Grassmannian in the regime $n>2k+1$, $k>1$. The obstruction is explicit: in $\mathrm{OGr}_+(2,6)$, two two-dimensional cells, one triangular and one square, have closures whose intersection is a triangle edge that is also a diagonal of the square, so the cell poset cannot be a regular CW complex. Alongside this, the paper establishes positive structural results: $\mathrm{OGr}_+(1,n)$ is combinatorially the product of simplices $\Delta^{\lceil n/2\rceil-1} \times \Delta^{\lfloor n/2\rfloor-1}$ and admits the canonical form given in Theorem 3.3, and $\mathrm{OGr}_+(k,2k+1)$ is linearly isomorphic to $\mathrm{OGr}_+(k+1,2k+2)$, with the faces of the former indexed by matchings on $[2k+2]$. The paper further defines orthopositroids---positroids for which the two sign sets $A^+_{IJ}$ and $A^-_{IJ}$ are simultaneously empty or nonempty---shows that every point of $\mathrm{OGr}_+(k,n)$ lies in an orthopositroid cell, and conjectures that every orthopositroid is realized by such a point.

Load-bearing premise

The general claim for all $k>1$ and $n>2k+1$ rests on the assertion that the displayed block-matrix extension embeds the $(2,6)$ triangle-square configuration into $\mathrm{OGr}_+(k,n)$ while preserving isotropy with respect to $\omega_0$, nonnegativity of all Plücker coordinates, and the offending gluing; the paper states this extension but does not prove that the appended $(k-2)\times(n-6)$ block satisfies the defining equations (5), so if that embedding fails, only the $k=2$, $n=6$ failure is established.

Editorial extensions

If this is right

For $n=2k+1$, the linear isomorphism between $\mathrm{OGr}_+(k,2k+1)$ and $\mathrm{OGr}_+(k+1,2k+2)$ reduces boundary computations to the known $n=2k$ case, where matchings on $[2k+2]$ index the cells.
For $k=1$, the boundary structure is a product of two simplices, and the canonical form from Theorem 3.3 gives an explicit logarithmic-form description of $\mathrm{OGr}_+(1,n)$ as a positive geometry.
For $n>2k+1$ and $k>1$, any future CW decomposition of $\mathrm{OGr}_+(k,n)$ cannot use the positroid cells as its cells; the paper's orthopositroid condition is the necessary first constraint on which positroid cells survive.
The Gröbner basis and primeness results imply that, for $n>2k$, the homogeneous coordinate ring of $\mathrm{OGr}(k,n)$ is described by the straightening-law quadrics and has the explicit degree from Proposition 2.7.
If Conjecture 5.3 holds, the orthopositroids of type $(k,n)$ give exactly the cells that meet $\mathrm{OGr}_+(k,n)$, providing the raw material for the missing cell decomposition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to compute the Euler characteristic of the cell complex assembled from the 99 orthopositroid cells of $\mathrm{OGr}_+(2,6)$ and compare it with the topological Euler characteristic of the variety; equality would support Conjecture 5.3, while failure would show that realizability alone does not guarantee a CW structure.
The triangle-square gluing suggests that the correct boundary complex for $n>2k+1$ may require cells that are not convex polytopes, or may require subdividing positroid cells, since a single positroid cell closure already exhibits non-regular incidence.
The same obstruction likely appears in the Plücker-positive flag variety $F_+(k,n)$, because $\mathrm{OGr}_+(k,n)$ is cut out from it by the diagonal condition; if so, a new cell decomposition for $F_+(k,n)$ in the $n>2k+1$ range would be a prerequisite for $\mathrm{OGr}_+(k,n)$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper has two solid sections and one load-bearing section that does not work as written. The stress-test note is correct. The displayed edge e1 in Section 5 has Plücker coordinate p35 = -b, so for b > 0 it is not in Gr+(2,6); if you fix the sign as the text seems to intend, that edge lies outside the closure of Cτ and cannot be one of its diagonals. Either way, the claimed triangle-square gluing is not established. Since this example is the only explicit obstruction to a CW decomposition and the template for the n > 2k+1 generalization, the paper's headline negative result is unsupported as it stands.

What is actually new and good: Section 3 gives a clean face decomposition of OGr+(1,n) as a product of simplices and provides a canonical form, and that part looks correct and useful. Section 4's isomorphism between OGr+(k,2k+1) and OGr+(k+1,2k+2) is a real result; the matchings-to-permutations dictionary is a nice contribution. The Gröbner basis material in Section 2 is mostly a repackaging of the prior spinor-helicity work [8], but the presentation is coherent and the connection to standard monomial theory is worth having.

Soft spots, in proportion: aside from the Section 5 failure, the proof of Theorem 4.5 handwaves at the crucial positivity step with "It is not so difficult to see." That is a minor exposition gap—the claim is plausibly true and easily checked—but it should be written out. More substantively, the enumeration of 99 orthopositroids in OGr+(2,6) is presented as an "exhaustive computation" with no code, data, or algorithmic description, so it is not independently reproducible. The block-matrix extension to all k,n in Section 5 also does not prove that isotropy with respect to ω0 and nonnegativity of all Plücker coordinates are preserved; it is a sketch. None of these would be fatal if Section 5's central example were valid, but it is not.

Who gets value: readers working on the positive orthogonal Grassmannian, positive geometries, or the combinatorics of Gr+(k,n). Sections 3 and 4 deserve a serious referee. The negative result does not, as written.

Recommendation: send it to peer review, but the referee should demand a repaired Section 5—either a corrected obstruction that actually lies in the positive region, or a downgrade of the CW claim to a conjecture. If the example cannot be fixed, the paper's frame changes substantially.

Referee Report

4 major / 5 minor

Summary. This paper studies the positive orthogonal Grassmannian OGr_+(k,n) for the alternating quadratic form ω0. The authors give a Gröbner basis and degree formula for the orthogonal Grassmannian OGr(k,n), describe OGr_+(1,n) as a product of simplices and as a positive geometry, prove an isomorphism between OGr_+(k,2k+1) and OGr_+(k+1,2k+2) with a matching description, and claim that for n>2k+1 and k>1 the positroid cells of Gr_+(k,n) do not induce a CW cell decomposition of OGr_+(k,n). The negative claim is supported by an example in OGr_+(2,6) involving a triangular cell and a square cell whose edge is claimed to glue to a diagonal.

Significance. If fully correct, the paper would make several useful contributions: a concrete positive-geometry structure for OGr_+(1,n), a bridge between OGr_+(k,2k+1) and OGr_+(k+1,2k+2), and a motivation for the new notion of orthopositroid. The exhaustive enumeration of 99 realizable orthopositroids in Table 1 and the representation-theoretic degree computation are valuable. However, the central negative theorem is not established as written: the only explicit obstruction in Section 5 is invalid, and the generalization to all k,n is only asserted. The remaining results may survive revision, but the main novelty needs a corrected argument or a replacement counterexample.

major comments (4)

[§5, displayed matrices after Mσ and Mτ] The edge e1 is not a boundary edge of the closure of Cσ in the positive orthogonal Grassmannian. For b>0, the Plücker coordinate p35 = det[[0,b],[1,0]] = -b is negative while p13 = 1, so the row span is not in Gr_+(2,6); for b=0, e1 is just the single vertex also lying on e3. If one instead takes the actual y=0 boundary of Cσ, represented by [[1,1,0,0,-x,-x],[0,0,1,1,0,0]] with x>0, then p35=p36=p45=p46=x>0, while every point of Cτ has p35=p36=p45=p46=0 identically, so this boundary is not in the closure of Cτ. The claimed gluing of a triangle edge to a diagonal of the square is therefore not established, and the only explicit obstruction to a CW decomposition in the paper disappears.
[§5, final paragraph starting 'In general this problem arises...'] The extension from OGr_+(2,6) to arbitrary (k,n) with n>2k+1 is asserted without proof. The displayed block extension is not shown to satisfy the orthogonality equations (5), nor is it shown that the embedded copies of Cσ and Cτ remain positive and preserve the claimed incidence. Even if the k=2,n=6 obstruction were valid, this paragraph would not establish the theorem for all k>1 and n>2k+1.
[§4, proof of Theorem 4.5] The positivity statement is justified by 'It is not so difficult to see' without a verification. Since the claim depends on the sign conventions in equation (5) and on which connected component is selected in OGr_+(k+1,2k+2), please provide an explicit check that the linear isomorphism sends the positive locus to the positive locus, or give a precise reference that contains this statement.
[§2, proof of Theorem 2.6] The passage from a Gröbner basis of I_{k,n,0} to one of I_{k,n} under the substitution φ is not justified. The map φ identifies q_J with ±p_{[n]\J}, so it is not injective on monomials: q_J and p_{[n]\J} have the same image, and cancellations can occur. The claim that an inequality between monomials 'continues to hold' after applying φ requires proof; please supply a direct argument or a standard reference showing that the images of the initial monomials generate in(I_{k,n}).

minor comments (5)

[§2, equation numbering] The text refers to '(8) and (8)' where two different displayed equations are evidently intended; please renumber them.
[§5, block matrix display] The block matrix used for the claimed generalization is difficult to parse; please rewrite it with clearly labeled blocks and explicitly state which entries are zero, which are free parameters, and how the blocks are chosen to satisfy the isotropy equations.
[§3, proof of Theorem 3.3] The proof of the positive-geometry property is abbreviated; in particular, the verification on all boundary strata and the treatment of the u_n=0 boundary are only sketched. Please expand the residue computations or indicate which steps are standard.
[Table 1] The enumeration of the 99 orthopositroids is presented as the output of 'an exhaustive computation'; please specify the computational method used and state whether code or a verification script is available.
[References] Reference [6] lacks journal or preprint data; please complete the bibliographic information. Also, 'A exhaustive computation' on page 20 should be 'An exhaustive computation'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivations are self-contained, and the self-citations point to external theorems that are not the target results.

full rationale

The paper's central claims are derived from explicit equations and constructions rather than from fitted parameters or definitions that presuppose the conclusions. The boundary structure of OGr+(1,n) is proved directly from the quadric equation and the parametrization (14); the positive-geometry statement in Theorem 3.3 is checked by explicit residue computations on the displayed canonical form. The isomorphism between OGr+(k,2k+1) and OGr+(k+1,2k+2) is supported by an explicit map (18) and by the coordinate-ring proof in Proposition 4.3, which is included in the text rather than assumed. The non-CW claim for OGr+(2,6) rests on the two explicit matrices M_sigma and M_tau and on the asserted triangle-square gluing; whether that gluing is geometrically correct is a question of validity, not circularity. The generalization to n > 2k+1 uses a block-matrix extension whose isotropy and positivity are not proved in detail, but that is an omitted verification, not an argument that assumes the conclusion. The only self-citation of note is [8, Theorem 2.7], used in Theorem 2.6 to transfer a Grobner basis from spinor-helicity varieties; that prior result is external to the paper's main boundary-cell claims and is not defined in terms of them. Conjecture 5.3 is explicitly left open, so it cannot serve as a hidden input. No equation in the paper is defined so that the target result becomes true by construction, and no fitted quantity is renamed as a prediction. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper is a theorem-based preprint; it fits no empirical constants, so the free-parameter list is empty. The main burden is carried by standard representation theory and by a Gröbner-basis theorem from a prior preprint by one of the authors, plus the fixed choice of the sign-alternating quadratic form. The one new combinatorial object, the orthopositroid, is introduced with a realizability conjecture rather than independent evidence.

assumptions (4)

standard math The prime ideal of OGr(k,n) is generated in degree 2 for n>2k, by Kostant's theorem.
Used in the proof of Theorem 2.9 to reduce the prime ideal statement to a degree-2 dimension count.
standard math Borel-Weil-Bott and the Weyl dimension formula compute the Hilbert function of the coordinate ring of OGr(k,n).
Used to prove the degree formula in Proposition 2.7 and the dimension check in Theorem 2.9.
domain assumption The Gröbner basis for the spinor-helicity ideal SH(k,n,0) from [8, Theorem 2.7] specializes to a Gröbner basis of I_{k,n} under the map sending q_{J'} to a signed complementary p-coordinate.
This is a cited prior theorem involving one of the present authors; it supports Theorem 2.6, but the paper's positive-geometry claims do not depend on this step.
domain assumption The sign-alternating quadratic form omega_0 in Eq. (1) is fixed, and positivity depends critically on the signature.
For other signatures the positive locus can be empty or zero-dimensional, as shown in Example 1.2. All subsequent statements are for this form.

invented entities (1)

Orthopositroid
purpose: A combinatorial condition on positroids meant to characterize cells of OGr_+(k,n) that occur in the intersection with the orthogonal quadrics.
Defined in Definition 5.1 from the sign pattern of equations (5). The paper verifies realizability for OGr_+(2,6) computationally and conjectures it in general in Conjecture 5.3. It is not used to derive the non-CW result, so this entry records a definitional burden rather than a fitted assumption.

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Pith. "Pith review of The positive orthogonal Grassmannian." pith.science (2026). https://pith.science/paper/U2QZGCKX

@misc{pith2026241214091,
  author       = {Pith},
  title        = {Pith review of: The positive orthogonal Grassmannian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2QZGCKX}},
  note         = {Machine review of arXiv:2412.14091}
}

abstract

The Pl\"ucker positive region $\mathrm{OGr}_+(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\mathrm{OGr}_+(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\mathrm{OGr}_+(1,n)$ in $\mathbb{P}^{n-1}_{+}$ and show that it is a positive geometry. We show that $\mathrm{OGr}_+(k,2k+1)$ is isomorphic to $\mathrm{OGr}_+(k+1, 2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_+(k,n)$.

Figures

Figures reproduced from arXiv: 2412.14091 by the authors.

**Figure 1.** The poset P2,6 is created from Y2,6 and Ye2,6 by adding the six covering relations in red. An incomparable pair of elements in Pk,n is of type ⟨i1,...,ik⟩,⟨j1,..., jk⟩ or ⟨i1,...,ik⟩,[ j ′ 1 ,..., j ′ n−k ] . Such a pair yields a non-semistandard Young tableau µ of shape (k, k) or λ of shape (n−k, k): µ = j1 ··· jℓ−1 j ℓ j ℓ+1 ··· j k i1 ··· iℓ−1 iℓ iℓ+1 ··· ik , λ = j ′ 1 ··· j ′ ℓ−1 j ′ ℓ j ′ ℓ+1 ··· j ′… view at source ↗

**Figure 2.** A lattice path depiction of the bijection in Lemma [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** The Hasse diagram of the poset structure on [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The positive Grassmannian OGr+ (1,4) is the red region in the tetrahedron P 3 +. The boundaries of OGr+(1,4) lie on the facets of P 3 +. 4. The positive Orthogonal Grassmannian OGr+(k,2k +1) We recall that we are working with the sign alternating form (1). The positro…

**Figure 5.** Figure 5: The face poset of OGr+(2,5) matches that of OGr+(3,6). See [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: A matching τ of [2k + 2] and the corresponding permutation σ of [2k+1] for k = 7. In the left figure, starting vertex 16 (in blue), the chords in red are longest sequence of chords c1,..., cr that intersect each other pairwise. In the figure on the right, the blue vert…

**Figure 7.** Figure 7: A cartoon of the cell Cσ (in green) glued to the cell Cτ (in red). This shows that the positroid cells are not enough to induce a CW cell decomposition on OGr+(2,6). In general this problem arises as soon as n > 2k+1. This is because whenever n > 2k+1 we have n−6 ≥ 2(…

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