Totally positive skew-symmetric matrices

Jonathan Boretsky; Veronica Calvo Cortes; Yassine El Maazouz

REVIEW 1 major objections 4 minor 43 references

Totally positive skew-symmetric matrices

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Pith's one-line read Total positivity of a real skew-symmetric matrix is exactly equivalent to positivity of $n(n-1)/2$ particular signed minors, and this is the fewest inequalities any such test can use.

desk verdict A real, carefully proved positivity criterion for skew-symmetric matrices, with one under-supported minimality claim that should be fixed before publication. read the letter →

arxiv 2412.17233 v1 pith:VBFZ6S7J submitted 2024-12-23 math.CO math.AG

Jonathan Boretsky , Veronica Calvo Cortes , Yassine El Maazouz This is my paper · ORCID

classification math.COmath.AG MSC 14M1515B4805E14

keywords orthogonalGrassmanniantotalpositivityskew-symmetricmatricesPfaffianssignedminorsMarsh-RietschparametrizationLindström-Gessel-ViennotdiagramsRichardsoncells

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 defines total positivity for real skew-symmetric matrices by viewing them as points of the totally positive orthogonal Grassmannian, and proves that a single explicit collection of $n(n-1)/2$ signed minors decides total positivity. The test is minimal: any positivity test by regular functions needs at least that many inequalities. The same geometric setup yields a Taylor-coefficient test for total nonnegativity, a fixed sign pattern for all Pfaffians of a totally positive matrix, and a matroid-based way to locate a matrix in the Richardson cell decomposition. If correct, total positivity for skew-symmetric matrices becomes a finite, checkable condition rather than a statement about all minors.

What carries the argument

The load-bearing object is the family of signed minors $M_{j,k}$, each a left-justified minor of the skew-symmetric matrix up to the sign $(-1)^{jk}$. The argument's engine is the Lindström-Gessel-Viennot diagram attached to a fixed reduced expression of the longest Weyl-group element: it turns each $M_{j,k}$ into the weight of a unique non-intersecting path collection, hence a signed monomial in the Marsh-Rietsch parameters, with exponent vectors forming a triangular system that lets the sign of $t_{\#(j,k)}$ be isolated from earlier minors. For the Pfaffian sign pattern, the machinery is the half-spin representation, where each Pfaffian appears, up to positive constants, as a generalized minor, so Lusztig positivity forces the intrinsic sign.

What would settle it

For $n=4$, compute the six minors $M_{j,k}$ from Definition 1.2 for a matrix $A(t_1,\dots,t_6)$ whose parameters in the fixed reduced expression (11) are chosen with mixed signs; if all six minors are positive while some $t_i$ is negative, Theorem 1.4 is false. The same check can be run for larger $n$ by searching the real parameter space with signed parameters.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that total positivity of $A \in S_n$ is equivalent to positivity of the $N = n(n-1)/2$ signed minors $M_{j,k}(A)$ for $1 \leq j \leq k \leq n-1$, where $M_{j,k}$ is the signed minor with rows $\{1,\dots,n-j\}$ and columns $\{1,\dots,n-k-1,n-k+j,\dots,n\}$, multiplied by $(-1)^{jk}$. The proof runs through the Marsh-Rietsch parametrization of the totally positive orthogonal Grassmannian: each $M_{j,k}$ is a signed monomial in the parameters $t_1,\dots,t_N$, and the signs of the parameters can be recovered one by one from the signs of the minors in reverse lexicographic order. The theorem also states that $N$ inequalities are the fewest possible for any regular-function positivity test. Beyond this, the paper shows that Pfaffians of a totally positive skew-symmetric matrix have the fixed sign pattern $\operatorname{sgn}(I,[n]) = (-1)^{\sum_{i \in I} i - \binom{|I|+1}{2}}$, and that the totally nonnegative part decomposes into positive Richardson cells whose identity can be read from the matroid of $[\mathrm{Id}_n \mid A]$.

Load-bearing premise

The proof that positivity of all the selected minors forces total positivity rests on the claim that each $M_{j,k}$ is, up to sign, a monomial in the Marsh-Rietsch parameters and that the sign of each newly introduced parameter can be separated from the signs of earlier minors in reverse lexicographic order; if that triangular structure or the sign cancellation failed, the criterion could stop being sufficient.

Editorial extensions

If this is right

Total positivity of skew-symmetric matrices is certified by exactly $N = n(n-1)/2$ explicit polynomial inequalities, and no regular-function test can use fewer.
The selected minors alone cannot detect total nonnegativity, but perturbing a matrix by a totally positive one-parameter family into $SO_{>0}(2n)$ and reading leading Taylor coefficients of the same minors gives a nonnegativity test.
Every totally positive skew-symmetric matrix has Pfaffians with the explicit sign $\operatorname{sgn}(I,[n])$; conversely, Pfaffian-positivity is strictly weaker and holds for matrices that are not totally nonnegative.
The totally nonnegative part $S_{\geq 0}^n$ is a disjoint union of positive Richardson cells, and the cell containing a matrix $A$ is determined by the matroid of $[\mathrm{Id}_n \mid A]$ through a lowering algorithm.
Because the parameter signs are Laurent monomials in the minors, the sign pattern of all $M_{j,k}$ completely determines $A$ among matrices with all $M_{j,k}$ nonzero.

Reading between the lines

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

One could turn the criterion into a practical certification routine: evaluate the $n(n-1)/2$ signed minors exactly or with interval arithmetic to certify total positivity without checking any other minor.
The triangular monomial structure suggests that the minors $M_{j,k}$ may form a cluster in some signed cluster structure on the orthogonal Grassmannian; the paper raises this possibility but does not prove it.
The strict inclusion $S_{\geq 0}^n \subset SPf_{\geq 0}^n$ points to a separate hierarchy of positivity cones for skew-symmetric matrices; analyzing the cell structure of the Pfaffian-positive cone would need tools beyond Weyl-group combinatorics.
A natural testable extension is to use the matroid-lowering algorithm to enumerate the Richardson cells for small $n$ and compare the resulting stratification with a direct sampling of random skew-symmetric matrices.

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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

Read it. The paper gives the first explicit total-positivity test for skew-symmetric matrices in the orthogonal Grassmannian sense: n(n-1)/2 signed minors M_{j,k} decide S>0_n, with a proof via Marsh-Rietsch parameters and LGV diagrams. That is a real result, and it is new. The same machinery yields a Pfaffian sign pattern (Theorem 1.7) and a matroid-based algorithm for locating the Richardson cell of a point (Theorem 4.25). The LGV-diagram part is worked out in detail; Lemmas 3.4, 3.7, and 3.8 carry the weight, and the monomial/triangularity argument looks sound to me. The Macaulay2 code is a plus, and the n=4 example in Example 2.3 is transparent.

The soft spot is exactly the one the stress-test note names: the minimality assertion in the second sentence of Theorem 1.4. The proof says any regular-function test pulls back to N polynomial inequalities defining the positive orthant, and then asserts without proof or citation that the orthant needs N inequalities. That is plausible and probably true, but it is not a one-line triviality; one has to handle redundant inequalities, zero gradients, and non-strict boundary behavior. As written, that part is a sketch. The main criterion does not depend on minimality, so this is a local fix: add a lemma with a proof, or cite a real-algebraic geometry source.

On Theorem 1.5, I am less worried than the reader. The equivalence (1) iff (2) is just the closure definition plus Lemma 4.1; (2) iff (3) follows from Theorem 1.4. The closing step where X(epsilon) tends to X and X lies in the closure is fine. What could be tightened is the uniform choice of the family Z(epsilon), but that is minor.

Citation pattern is honest: Lusztig and Marsh-Rietsch are the framework, and the self-citations to [9] and [10] are for background and code, not load-bearing. No sign of circularity.

The audience is people working in total positivity, positroid theory, and the combinatorial side of the orthogonal Grassmannian; they will get real value from the explicit criterion and the cell-membership algorithm. Bottom line: this is a useful, carefully written paper. The main theorem deserves a serious referee. The minimality claim needs a proper proof or a reference; without it, the second sentence of Theorem 1.4 should be flagged as conditional, not removed. I would send it to peer review with a request for that revision.

Referee Report

1 major / 4 minor

Summary. The paper studies the space S_n of real skew-symmetric n×n matrices as an affine chart of the orthogonal Grassmannian OGr(n,2n). Total positivity is defined through Lusztig's positive part, using the Marsh–Rietsch parametrization relative to a fixed reduced expression. The central result, Theorem 1.4, asserts that A∈S_n is totally positive if and only if the N=n(n−1)/2 signed minors M_{j,k}(A) of Definition 1.2 are all positive, and that this test is minimal in the number of inequalities used. The proof is built from a monomial analysis in the LGV diagram: Lemma 3.4 shows the M_{j,k} are monomials in the Marsh–Rietsch parameters and that the parameters can be recovered in reverse lexicographic order, Lemma 3.7 handles the signs, and Lemma 3.8 uses a density argument to extend the recovery from the parametrized set to all matrices with nonzero M_{j,k}. The paper also proves a nonnegativity test via limits of totally positive deformations (Theorem 1.5), a matroid-based determination of the Richardson cell containing a given point (Theorem 1.6 and Theorem 4.25), and a fixed sign pattern for Pfaffians (Theorem 1.7).

Significance. If it stands, the main equivalence gives an explicit and practical positivity criterion: total positivity of a skew-symmetric matrix is checked by N honest minor inequalities, exactly mirroring the classical type-A picture. The monomial/triangularity analysis in Lemmas 3.4 and 3.7 is detailed and internally consistent, and I found no gap in the central derivation. The matroid-based cell identification of Theorem 4.25 is a further genuine contribution toward orthogonal positroids, and the Pfaffian sign theorem is elegant. The manuscript also ships Macaulay2 code backing the examples, which is a reproducible-checking strength. The main weakness is the minimality assertion in Theorem 1.4: the lower bound on which it rests is stated without proof or reference, so the advertised minimality is not established, although the 'if and only if' criterion itself appears sound.

major comments (1)

[Theorem 1.4 (second sentence), proof at end of Section 3] The minimality assertion is not proved. The proof reduces any positivity test by regular functions to r polynomial inequalities cutting out the positive orthant (R_{>0})^N, and then invokes the statement that the orthant in R^N cannot be cut out by fewer than N polynomial inequalities. No proof or citation is supplied. This lower bound is nontrivial: the case r=1 can be handled by showing a polynomial positive on the orthant and nonpositive on its complement must vanish on every coordinate hyperplane, but the general case requires a genuine argument, and the scope of the claim should be stated explicitly (e.g., 'among tests that are conjunctions of strict inequalities by regular functions'). Without a proof or reference, the second sentence of Theorem 1.4 is unsupported, even though the first sentence—the positivity criterion itself—is unaffected. The theorem statement, the abstract, and the introduction advertise the test as minimal, so this is a load-bearing point that needs to be fixed.

minor comments (4)

[Proposition 4.27] The statement as printed is incomplete: the displayed union is over v∈W_{[n−1]} only, and the symbol w is not introduced. It should be a union over all minimal coset representatives w and all v≤w, consistent with equation (4). This is likely a typographical omission, but it should be corrected.
[Lemma 3.8] The rational map φ from U to V is defined by 'determining the t_i as described in the proof of Lemma 3.4', but Lemma 3.4 is stated and proved for points already in the parametrized set V. The intended argument is valid: one first uses the Laurent-monomial formulas to define a rational map on U, observes that it restricts to the identity on V, and then uses density to conclude. Making this two-step definition explicit would improve readability and remove a perceived circularity.
[Conventions around Definition 1.2] The notation M_{j,k} is used both for the signed minors and, inside the proof of Lemma 3.4, for the left-greedy monomial minor; the unsigned variant is introduced but the switch is easy to miss. A brief note near the start of Section 3 would help.
[Theorem 1.5] The theorem states the equivalence for any smooth one-parameter family Z(ε) in SO_{>0}(2n) tending to the identity, and then adds that Z(ε) can be chosen so that the minors M_{j,k}(B(ε)) are polynomials in ε. The proof is clear, but the hypothesis that the family be smooth enough to possess Taylor expansions should be stated up front, since condition (3) is phrased in terms of leading coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.4 is a proved coordinate-change equivalence; cited self-work is not load-bearing.

full rationale

The central derivation of Theorem 1.4 is self-contained in the relevant sense: total positivity is defined through the Marsh-Rietsch parametrization (Definition 2.2 and equation (13)), and the criterion M_{j,k}(A)>0 is proved rather than assumed. Lemma 3.4 shows that each M_{j,k} is a monomial in the parameters t_i and that the sign of each t_i can be recovered from the signs of the M_{j,k} by reverse induction; Lemma 3.7 computes the monomial signs; Lemma 3.8 uses Zariski density to show that the M_{j,k} determine the matrix. Neither quantity is defined as the other, so the equivalence is a coordinate-change theorem rather than an identity. The parametrization [35, Theorem 11.3] and the positivity projection results [32,33] are external to this paper's authors; self-citations to [9] (an LGV corollary) and [10] (Macaulay2 code) support exposition and computation and are not load-bearing for the main equivalence. The only flagged weakness is non-circular: the minimality sentence in Theorem 1.4 rests on the asserted fact that (R_{>0})^N cannot be cut out by fewer than N polynomial inequalities, stated without proof at the end of Section 3; this is an unsupported lower-bound claim, not a reduction of the criterion to its inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's contribution is a new coordinate criterion, not a new postulate: it uses standard total positivity machinery as input. The only nonstandard auxiliary assertion is the orthant lower bound used for the minimality claim.

assumptions (8)

standard math Definition and properties of the totally positive part of a reductive group and its flag varieties in the sense of Lusztig, including projection from complete flags to partial flags.
Used to define S>0_n in Definition 1.1 and to write the parametrization in equations (10)-(13).
standard math Marsh-Rietsch parametrization of the totally positive complete flag variety and of its Deodhar components, Theorem 11.3 and Section 5 of [35].
Gives the coordinate system (t_1,...,t_N) on which the proof of Theorem 1.4 is built.
standard math Lindström-Gessel-Viennot lemma as applied to concatenations of directed weighted graphs, Corollary 2.21 of [9] and also [21,31].
Used in Proposition 2.8 and Proposition 4.13 to express maximal minors as sums over non-intersecting path collections.
standard math Semigroup property: for Z in SO>0(2n) and X in OGr≥0(n,2n), X·Z is in OGr>0(n,2n), citing [32, Proposition 8.17] and [33, Theorem 3.4].
Used in Lemma 4.1 and essential for the nonnegativity test Theorem 1.5.
standard math Expression of minors of a skew-symmetric matrix in terms of its Pfaffians, from [14, Theorem 1].
Used in Lemma 3.5 to express the diagonal minors M_{k,k} as ratios of Pfaffians.
standard math Every rank-k matroid has a unique Gale-maximal basis with respect to any total order on the ground set, [19, Theorem 4].
Used in Section 4.5 to define the base basis I_X and the lowering sequence in Theorem 4.25.
standard math The Richardson/Deodhar decomposition of the totally nonnegative part of a flag variety, with positive Richardson cells indexed by pairs (v,w) in Bruhat order, [28,35].
The cell decomposition S≥0_n = union of positive Richardson cells in equation (4) relies on this.
ad hoc to paper The positive orthant (R>0)^N cannot be defined by fewer than N polynomial inequalities.
Asserted at the end of the proof of Theorem 1.4 without proof or reference; it supports the minimality claim.

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Pith. "Pith review of Totally positive skew-symmetric matrices." pith.science (2026). https://pith.science/paper/VBFZ6S7J

@misc{pith2026241217233,
  author       = {Pith},
  title        = {Pith review of: Totally positive skew-symmetric matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBFZ6S7J}},
  note         = {Machine review of arXiv:2412.17233}
}

abstract

A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian $\mathrm{OGr}(n,2n)$. Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid.

Figures

Figures reproduced from arXiv: 2412.17233 by the authors.

**Figure 2.** The collection of non-intersecting paths in the LGV diagram corresponding to the minor M2,1(A) for n = 5. Theorem 1.4. A skew-symmetric matrix A ∈ Sn is totally positive if and only if Mj,k(A) > 0 for any 1 ≤ j ≤ k ≤ n − 1. This test is minimal in the sense that it uses the fewest possible number of inequalities. The set S ≥0 n of totally nonnegative skew-symmetric matrices is the Euclidean closure of S >0 n . Whil… view at source ↗

**Figure 3.** The unique path collection from I = {1, 2, 3, 4, 5} to {3, 4, 6, 7, 8}. Proposition 2.8. Let I ∈ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Illustration of the path collection corresponding to M1,1. For k = 2, j = 1 we modify the path starting at 1 and obtain the following path collections corresponding to M1,2 = (t1t2t3t4) 2 t5t6 for n = 4 and M1,2 = (t1 · · ·t8) 2 t9t10 for n = 5 [PITH_FULL_IMAGE:figure…

**Figure 5.** Figure 5: Illustration of the path collection corresponding to M1,2. The path collection in example 2.7 is obtained from the previous one by changing the path starting at 2 from left greedy to right greedy, and it corresponds to M2,2 = (t1 · · ·t7) 2 t8t9 for n = 5. Proof of Pro…

Discussion (0). Continue with ORCID to comment.

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work page 2019

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Mathrepo code repository for totally pos- itive skew symmetric matrices

Boretsky, J., Calvo Cortes, V., and El Maazouz, Y. Mathrepo code repository for totally pos- itive skew symmetric matrices. Available at https://mathrepo.mis.mpg.de/PositiveSkewMatrices

work page

[3] [1]

jump arrows

By Lemma 3.1, the left greedy path collection originating from [ n] unique and uses all arrows in the LGV diagram, so M = (t1 · · · tN )2. For k = 1, j = 1, the unique path collection ( n − 1, 2n − 1, . . . , n+ 1) is almost identical to the left greedy path collection originating from [ n], only the path originating at 1 is modiﬁed. Recall from the proof...

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[4] [2]

Positive geometries and canonical forms

Arkani-Hamed, N., Bai, Y., and Lam, T. Positive geometries and canonical forms. J. High Energy Phys. 2017 , 11 (2017), 039, front matter+121

work page 2017

[5] [3]

backwards

and ( 4). The paths in this path collection are all left greedy, proving ( 3). We next show that for k ∈ [n−1] the path originating from source vertex k terminates below the path originating from source vertex k +1. The only place where this might break is if the left greedy paths pa ths in the truncation of Gu,w in (i) originating at k and k + 1 terminat...

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[6] [4]

Each path is directed from i to w−1(i) for some i ∈ [n]

is satisﬁed as well. Each path is directed from i to w−1(i) for some i ∈ [n]. Since w ∈ W [n−1], by Remark 4.16, w−1(1) ≺ · · · ≺ w−1(n), proving (2). In this base case, ( 1) does not assert anything. Finally, the next lemma implies it is a unique path collection. Lemma 4.20. Suppose Gv,w has all vertical edges pointing upwards. Let P be a non- intersecti...

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[7] [5]

Positively oriented matroids are realizable

Ardila, F., Rinc ´on, F., and Williams, L. Positively oriented matroids are realizable. J. Eur. Math. Soc. (JEMS) 19 , 3 (2017), 815–833

work page 2017

[8] [6]

On two notions of total positivity for generalized partial flag varieties of classical Lie types

Barkley, G., Boretsky, J., Eur, C., and Gao, J. On two notions of total positivity for generalized partial ﬂag varieties of classical lie types. arxiv:2410.11804 (2024)

work page Pith review arXiv 2024