{"id":"5471dbf6-2e40-4dbc-b7f4-84fe69531699","arxiv_id":"2412.17233","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An n×n skew-symmetric matrix is totally positive exactly when its n(n−1)/2 selected signed minors are positive, and its Pfaffians then follow one universal sign rule.","lead":"Skew-symmetric matrices can never have all positive entries, so the authors define a different notion of total positivity for them using the orthogonal Grassmannian. They identify a short list of signed minors whose positivity exactly detects this property, and show that Pfaffians on such matrices obey a fixed sign pattern.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's positivity criterion is well supported; the minimality assertion rests on an unproved orthant lower bound.","rationale":"I read the central equivalence as the core of the paper. The Marsh–Rietsch parametrization, the unique LGV path collections in Proposition 3.2, and the triangular recovery of the parameters in Lemma 3.4 together give a credible proof of the iff criterion; Lemma 3.7's sign count checks out, and Lemma 3.8's rational-map argument is valid because the parametrization image is Zariski dense. I could not identify a concrete flaw in this part, contrary to the reader's weakest_assumption. The true soft spot is the minimality statement: the paper asserts the orthant lower bound without proof. This is a real gap because Theorem 1.4 explicitly advertises minimality, and the supplied one-sentence reduction to R^N does not itself establish the lower bound. The reader's verdict of CONDITIONAL is therefore appropriate, and the recommended revision should add a proof or reference for the orthant lower bound. My agreement with the reader is partial: the reader's weakest_assumption names the monomial/triangularity step, which I find solid, while their rationale already flags the minimality argument, which I agree is the main caveat.","tokens_in":40573,"tokens_out":22465,"duration_ms":205394,"concrete_test":"Settle the orthant lower bound: prove that for N≥2 there is no polynomial p∈R[t_1,...,t_N] with {p>0}=R_{>0}^N. Concretely, run real quantifier elimination (Mathematica Reduce or verified CAD) for N=2, degree ≤4, searching for such a p; then complete the general proof by showing any such p vanishes on the positive part of each hyperplane t_i=0, forcing divisibility by every t_i and an infinite descent. If the proof goes through, the minimality argument in Theorem 1.4 is filled in; if a counterexample appears, the minimality assertion must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main equivalence in Theorem 1.4 is carefully argued through Lemmas 3.4, 3.7, and 3.8; I found no internal gap in the monomial/triangularity analysis or the sign bookkeeping. The load-bearing weakness is the minimality claim in the second sentence of Theorem 1.4. Its proof at the end of Section 3 asserts that any positivity test using r regular functions pulls back to r polynomial inequalities cutting out the positive orthant (R>0)^N, and that the orthant in R^N cannot be cut out by fewer than N polynomial inequalities. No proof or citation is supplied for this nontrivial lower bound. The coordinate inequalities t_i>0 give N inequalities, but it is not obvious that no smaller system of strict polynomial inequalities defines the orthant; a standard proof would show such a polynomial must vanish on every coordinate hyperplane in the closure and hence be divisible by each t_i, producing an infinite descent. If this lower bound failed, or if tests with disjunctions or non-regular functions were allowed, the minimality assertion would collapse. The main 'if and only if' criterion would survive, but Theorem 1.4 as stated would need revision.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":40781,"tokens_out":18448,"duration_ms":177643,"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":[{"comment":"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.","section":"Theorem 1.4 (second sentence), proof at end of Section 3"}],"minor_comments":[{"comment":"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.","section":"Proposition 4.27"},{"comment":"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.","section":"Lemma 3.8"},{"comment":"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.","section":"Conventions around Definition 1.2"},{"comment":"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.","section":"Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to total positivity in type D, and the main equivalence theorem is well supported. The only load-bearing defect is the unproved minimality lower bound in Theorem 1.4; this is local and fixable either by supplying a proof or a reference, or by qualifying the minimality claim. I recommend major revision rather than rejection, since the central criterion and the cell-identification results appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nOn 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.\n\nCitation 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.\n\nThe 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.","headline":"A real, carefully proved positivity criterion for skew-symmetric matrices, with one under-supported minimality claim that should be fixed before publication.","tokens_in":41320,"tokens_out":10458,"would_cite":true,"duration_ms":97390,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","15B48","05E14"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["orthogonal Grassmannian","total positivity","skew-symmetric matrices","Pfaffians","signed minors","Marsh-Rietsch parametrization","Lindström-Gessel-Viennot diagrams","Richardson cells"],"falsifier":"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.","tokens_in":40357,"feed_emoji":"🧮","tokens_out":7391,"duration_ms":59247,"temperature":0.7,"pith_summary":"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.","feed_headline":"n(n-1)/2 signed minors decide total positivity of skew matrices","feed_subtitle":"A finite, provably minimal set of inequalities replaces checking every minor of the skew-symmetric matrix.","key_machinery":"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.","core_discovery":"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 $[\\mathrm{Id}_n \\mid A]$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the Marsh-Rietsch parametrization of the totally positive flag variety used to write every point of $OGr_{>0}(n,2n)$ as $A(t_1,\\dots,t_N)$.","marker":"[35]"},{"why":"It defines total positivity in split reductive groups and gives the semigroup and projection facts behind the nonnegativity test.","marker":"[32]"},{"why":"It extends Lusztig positivity to partial flag manifolds, allowing projection from the complete flag and identifying Pfaffians as positive generalized minors.","marker":"[33]"},{"why":"It provides the LGV-diagram construction for maximal minors of flag matrices that the paper adapts to the orthogonal Grassmannian.","marker":"[9]"},{"why":"It is the Gessel-Viennot determinant lemma used in Proposition 2.8 to expand maximal minors as signed path sums.","marker":"[21]"},{"why":"It is Lindström's path lemma, invoked together with [21] for non-intersecting path expansions of determinants.","marker":"[31]"},{"why":"It gives minors of a skew-symmetric matrix in terms of Pfaffians, used in Lemma 3.5 to express the full monomial as a Laurent monomial in diagonal minors.","marker":"[14]"},{"why":"It provides the Richardson and Deodhar decomposition framework used to describe cells of the totally nonnegative orthogonal Grassmannian.","marker":"[28]"}],"fun_headline_variants":["Signed minors decide total positivity of skew-symmetric matrices","Minimal set of signed minors for total positivity","Fixed sign pattern for Pfaffians of totally positive skew matrices","Skew matrices: positivity from just n(n-1)/2 signed minors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Signed minors decide total positivity of skew-symmetric matrices","Minimal set of signed minors for total positivity","Fixed sign pattern for Pfaffians of totally positive skew matrices","Skew matrices: positivity from just n(n-1)/2 signed minors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001199,"raw_usage":{"total_tokens":4964,"prompt_tokens":985,"completion_tokens":3979,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":3910}},"tokens_in":601,"tokens_out":3979,"duration_ms":24314,"temperature":1.0,"reasoning_tokens":3910,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:41:27.451727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the vector representations of induced matroids","cited_arxiv_id":null,"evidence_quote":"It supplies the Marsh-Rietsch parametrization of the totally positive flag variety used to write every point of $OGr_{>0}(n,2n)$ as $A(t_1,\\dots,t_N)$."},{"cited_title":"The deodhar decomposition of the grassmannian and the regularity of KP solitons","cited_arxiv_id":null,"evidence_quote":"It defines total positivity in split reductive groups and gives the semigroup and projection facts behind the nonnegativity test."},{"cited_title":"Total positivity for cominuscule Grassmannians","cited_arxiv_id":null,"evidence_quote":"It extends Lusztig positivity to partial flag manifolds, allowing projection from the complete flag and identifying Pfaffians as positive generalized minors."},{"cited_title":"Matroids over partial hyperstructures","cited_arxiv_id":null,"evidence_quote":"It provides the LGV-diagram construction for maximal minors of flag matrices that the paper adapts to the orthogonal Grassmannian."},{"cited_title":"The vector space S can be endowed with an action of the spin group Spin(2 n)","cited_arxiv_id":null,"evidence_quote":"It is the Gessel-Viennot determinant lemma used in Proposition 2.8 to expand maximal minors as signed path sums."},{"cited_title":"Positroid Stratification of Orthogonal Grassmannian and ABJM Amp litudes","cited_arxiv_id":null,"evidence_quote":"It is Lindström's path lemma, invoked together with [21] for non-intersecting path expansions of determinants."},{"cited_title":"W., Veliche, O., and Weyman, J","cited_arxiv_id":null,"evidence_quote":"It gives minors of a skew-symmetric matrix in terms of Pfaffians, used in Lemma 3.5 to express the full monomial as a Laurent monomial in diagonal minors."},{"cited_title":"ABJM amplitudes and the positive orthogonal grassmannian","cited_arxiv_id":null,"evidence_quote":"It provides the Richardson and Deodhar decomposition framework used to describe cells of the totally nonnegative orthogonal Grassmannian."}],"review_version":1}