{"id":"1891e20b-8d32-4958-b560-8d50f264f3e0","arxiv_id":"1908.07373","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit formulas for Chern-Schwartz-MacPherson classes of all GL_n(C)-orbits in symmetric and skew-symmetric matrices, via a sieve method and an interpolation method.","lead":"The paper derives two explicit formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric matrix degeneracy loci. It uses them to compute Euler characteristics of linear sections and to point toward K-theory analogues.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The symmetric half of the central theorem is asserted without a proof in the paper: Theorem 5.5 refers to an unpublished thesis and Theorem 4.7 says the argument is analogous, so the full csm(Σ^S_{n,r}) = W^S_{n,r} claim is not self-contained.","rationale":"Reader's verdict already flags the same point, and I agree: the symmetric family is where the paper's full claim is least secure. The skew-symmetric proof is not obviously flawed; the W-functions and sieve formulas match the stated examples, and the degree and divisibility arguments are plausible. But the symmetric statements depend on the assertion that the argument is analogous, and the interpolation theorem is representation-specific, so the analogy is not guaranteed. In particular, the stabilizer, normal bundle, parity structure, and matrix inversion all differ from the skew case. The remedy is an independent verification of Theorem 3.4's conditions for W^S and an independent derivation of the Pascal inversion in Theorem 4.7. None of this is an accusation of error; it is a request for the missing proof. Therefore the appropriate verdict remains CONDITIONAL, unchanged from the reader.","tokens_in":21465,"tokens_out":16416,"duration_ms":149751,"concrete_test":"For n=3 and n=4, all r, compute W^S_{n,r} symbolically from Definition 5.3 and test the three interpolation conditions of Theorem 3.4 against the actual stabilizer data: restrict W^S to the O(n-r)×GL_r maximal torus and compare with c(TΣ^S_{n,r})e(NΣ^S_{n,r}); check divisibility by c(TΩ) after restricting to each lower orbit; and check the degree inequality. Independently, recompute Theorem 4.7 by writing the fibered resolution over Gr_r(C^k) for S^2 C^n, obtaining Φ^S_{n,r} = sum binom(k,r) ssm(orbit_k), and invert the Pascal matrix to confirm the stated (-1)^i binom(r+i,r) coefficients. If all checks pass, the symmetric formulas are likely correct and the gap is expositional; if any fail, the theorem needs modification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is the pair of identifications in Theorem 5.5. The skew-symmetric half is proven in detail. The symmetric half, csm(Σ^S_{n,r}) = W^S_{n,r}, is disposed of by the sentence 'The proof ... is analogous, we leave it to the reader (or see [P])', where [P] is an unpublished Ph.D. thesis. The same happens for the symmetric sieve formula: Theorem 4.7 is introduced with 'Arguments analogous to those in Section 4.2 give the following theorem, we leave the details to the reader.' This is a genuine load-bearing gap because the two representations are not parallel in the ways that matter for the interpolation theorem. The symmetric stabilizer is O(n-r,C) × GL_r(C), not Sp(n-r,C) × GL_r(C); the normal representation is S^2 of the kernel, not Λ^2; the orbit parity is unrestricted; and the sieve inversion involves the full Pascal matrix rather than the even Euler-number matrix. Consequently the three conditions of Theorem 3.4—the restriction identity φ_Σ(csm(Σ)) = c(TΣ)e(NΣ), the divisibility by c(TΩ) on every lower orbit, and the degree bound for property (3)—require separate verification. The text does not even record e(N) ≠ 0 for the symmetric stabilizer, though that condition is part of Assumption 3.3. Nothing in the paper indicates the symmetric formulas are false—the small examples in Section 6 and the normalization checks are consistent—but the submitted text leaves the full central claim dependent on an external, unpublished source.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the GL_n(C)-equivariant Chern-Schwartz-MacPherson (CSM) and Segre-Schwartz-MacPherson (SSM) classes of the orbits in the representations Λ^2 C^n and S^2 C^n, i.e., of skew-symmetric and symmetric degeneracy loci. It presents two families of formulas: a \"sieve formula\" obtained from fibered resolutions and an inversion of a binomial/Euler-number matrix (Theorems 4.5 and 4.7), and an \"interpolation formula\" expressing the CSM classes as explicit symmetric polynomials, the W-functions (Theorem 5.5). The skew-symmetric case is proven in detail: the W-function is verified against the three interpolation conditions of Theorem 3.4, with explicit stabilizer, tangent, and normal representations. The symmetric case is stated by analogy, with the proof left to the reader or to the unpublished thesis [P]. The paper then gives Schur and ˜s_lambda expansions, formulates positivity and symmetry conjectures, computes non-equivariant CSM classes after projectivization, and applies Aluffi's J-transform to obtain Euler characteristics of general linear sections.","tokens_in":21860,"tokens_out":5921,"duration_ms":64712,"significance":"If both halves of the main theorem hold, the paper provides explicit, parameter-free, and algorithmically usable formulas for characteristic classes of classical symmetric and skew-symmetric degeneracy loci, a natural CSM deformation of the Józefiak-Lascoux-Pragacz and Harris-Tu formulas. The skew-symmetric proof is detailed and self-contained modulo the published interpolation theorem [FR2]; the examples are concrete; and the application to Euler characteristics of linear sections is clearly explained. However, the full claim of the paper is not self-contained: the symmetric half of the central theorem is delegated to an unpublished source, and the symmetric sieve formula is asserted without proof. Since the symmetric and skew-symmetric settings differ in stabilizer, normal representation, orbit parity, and the matrix inversion step, this is a genuine load-bearing gap rather than a harmless presentational shortcut.","major_comments":[{"comment":"The proof of the second sentence of Theorem 5.5, `csmp(Σ^S_{n,r}) = W^S_{n,r}`, is not contained in the paper: the final paragraph says the argument is analogous and leaves it to the reader or to the unpublished thesis [P]. This is load-bearing because the symmetric case is not a formal paraphrase of the skew case: the stabilizer of a point in Σ^S_{n,r} is O(n−r,C)×GL_r(C) rather than Sp(n−r,C)×GL_r(C), the normal representation is S² of the kernel rather than Λ², all coranks 0≤r≤n occur, and the sieve inversion involves the full Pascal matrix rather than the even-submatrix with Euler numbers. The three conditions of Theorem 3.4 therefore require independent verification in the symmetric setting, including the divisibility condition (2) and the degree bound in condition (3). I recommend either including the full proof or providing a publicly available reference that contains it.","section":"§5.2, Theorem 5.5"},{"comment":"Theorem 4.7 is introduced with the sentence `Arguments analogous to those in Section 4.2 give the following theorem, we leave the details to the reader.` The theorem states the symmetric sieve formulas for both the open orbit and its closure, with no proof or explicit description of the relevant fibered resolution, the fiber Euler characteristics, or the matrix inversion. Since Theorem 4.7 is the symmetric counterpart of Theorem 4.5 and is central to the paper's full claim, this omission leaves a main result unsupported. Please supply the derivation, or at minimum state the symmetric fibered resolution and the invertible matrix explicitly before referring to [P].","section":"§4.3, Theorem 4.7"},{"comment":"For the skew case the proof explicitly checks Assumption 3.3 by computing e(N_{Σ^∧_{n,r}}) ≠ 0 and then verifies the three interpolation properties. No analogous check is recorded for S²C^n: the statement that W^S_{n,r} is a symmetric polynomial with the stated top degree is asserted without proof, and the nonvanishing of the Euler class of the normal representation is not verified. These facts are part of the hypotheses of Theorem 3.4 and of the degree comparison in condition (3), so they should be stated and proved for the symmetric representation before the analogous argument is invoked.","section":"§5.2 / Assumption 3.3"}],"minor_comments":[{"comment":"There is a typo, `bunddle` for `bundle`, in the first sentence of the proof.","section":"§4.1, proof of Proposition 4.1"},{"comment":"The two displayed formulas in Theorem 4.7 appear with the same symbol `Σ^S_{n,r}` for the orbit; if one formula is for the open orbit and the other for its closure, the notation must be distinguished.","section":"§4.3, Theorem 4.7"},{"comment":"In the paragraph proving property (3) for the r=0 case, the text says `Assume that 0 < m ≤ k`, but the variables in the surrounding argument are n and m; this should be corrected to avoid confusion.","section":"§5.2, degree estimate"},{"comment":"There is a typo, `obtian` for `obtain`, in the computation of φ_{Σ^∧_{n,0}}(W^∧_{n,0}).","section":"§5.2, r=0 verification"},{"comment":"The operation J and the relation with Aluffi's theorem are stated correctly, but the notation γ_X(t) is introduced before the relation χ_X ↔ γ_X is motivated; a sentence explaining that γ_X is the reversed coefficient polynomial would improve readability.","section":"§7.3"}],"recommendation":"major_revision","confidential_remarks":"The main structural issue is that a central theorem is delegated to the authors' own unpublished thesis [P]. I would advise asking the authors to include the symmetric proof in the paper or to make the thesis publicly available and to state precisely which arguments are quoted from it. The use of [FR2]'s interpolation theorem as a black box is appropriate and not circular. The skew-symmetric contribution is substantial and appears sound; the revision should focus on closing the symmetric-case gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something real: it gives explicit CSM-class formulas for the GL_n(C) orbits in both Λ^2 C^n and S^2 C^n, introduces new W-functions, and derives a sieve formula from explicit fibered resolutions. The skew-symmetric side is worked out in careful detail—the verification of the three interpolation properties in Theorem 5.5 is thorough and does not rely on any fitted parameters. That half is a genuine advance over the fundamental-class formulas of Jozefiak–Lascoux–Pragacz and Harris–Tu.\n\nThe applications are also handled honestly: Aluffi's involution converts the non-equivariant CSM classes into Euler characteristics of linear sections, and the small examples check out. The positivity conjectures are clearly labeled as conjectures, and the normalization checks (sum of SSM classes over all orbits equals 1) are a nice sanity test.\n\nThe soft spot is exactly where the stress-test note lands. The symmetric half of Theorem 5.5 is asserted with 'the proof is analogous, we leave it to the reader (or see [P])', where [P] is an unpublished PhD thesis. Theorem 4.7, the symmetric sieve formula, is likewise stated with details left to the reader. This is not a cosmetic omission. The symmetric stabilizer is O(n-r)×GL_r rather than Sp(n-r)×GL_r, the normal representation is S^2 of the kernel rather than Λ^2, the orbit parity is unrestricted, and the sieve inversion involves the full Pascal matrix rather than the even Euler-number matrix. So the three conditions of Theorem 3.4 require separate verification; the paper does not even record e(N) ≠ 0 for the symmetric stabilizer, which is part of Assumption 3.3. The formulas are probably correct—the examples and the normalization checks are consistent—but the submitted text is not self-contained on this point. The reader's CONDITIONAL verdict is appropriate.\n\nFor a reader who works in characteristic classes or enumerative geometry, the skew-symmetric results alone are worth a look. The paper deserves a serious referee, but the referee should insist that the symmetric proofs be written out or that the paper be trimmed to the skew-symmetric case. I would not cite the symmetric half in its current state, though I would gladly cite the skew-symmetric formulas once they stand alone.","headline":"A serious, technically solid extension of the FR2 interpolation framework with a genuine, load-bearing gap: the symmetric half of the main theorem is deferred to an unpublished thesis.","tokens_in":22354,"tokens_out":1712,"would_cite":true,"duration_ms":18928,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C17","14M12","14L30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes explicit finite formulas for the Chern-Schwartz-MacPherson classes of all orbits in the spaces of symmetric and skew-symmetric matrices under the general linear group, via a sieve formula and an interpolation formula.","keywords":["Chern-Schwartz-MacPherson classes","Segre-Schwartz-MacPherson classes","degeneracy loci","symmetric matrices","skew-symmetric matrices","equivariant cohomology","interpolation","motivic Chern classes"],"falsifier":"Compute $\\varphi_{\\Sigma^S_{n,r}}(W^S_{n,r})$ for a small case such as $(n,r)=(3,1)$ and compare with $c(T\\Sigma^S_{3,1})e(N\\Sigma^S_{3,1})$; any discrepancy would disprove Theorem 5.5. Independently, compare $W^S_{3,1}$ with the CSM class obtained from the resolution-sieve formula for $S^2\\mathbb{C}^3$.","tokens_in":21270,"feed_emoji":"🧮","tokens_out":10405,"duration_ms":92681,"temperature":0.7,"pith_summary":"This paper computes Chern-Schwartz-MacPherson (CSM) classes, the characteristic classes that deform the fundamental class and record topological Euler characteristics, for the orbit strata in $\\Lambda^2\\mathbb{C}^n$ and $S^2\\mathbb{C}^n$ under $GL_n(\\mathbb{C})$. It gives two descriptions: a sieve formula expressing Segre-Schwartz-MacPherson classes as finite alternating sums of explicitly computed $\\Phi$-classes, and an interpolation formula identifying each CSM class with a symmetric polynomial $W_{n,r}$ defined by a symmetric-group averaging expression. These formulas cover both the skew-symmetric and symmetric degeneracy loci for all ranks $r$, and they are applied to Euler characteristics of general linear sections and to positivity and K-theoretic questions.","feed_headline":"Explicit Chern-class formulas for symmetric and skew-symmetric loci","feed_subtitle":"Equivariant CSM classes of matrix orbits are identified with explicit W-functions and alternating sums.","key_machinery":"The load-bearing objects are the $W$-functions $W^\\wedge_{n,r}$ and $W^S_{n,r}$: averages over permutations and sums over $r$-subsets of Chern roots $\\alpha_i$ of products like $(\\alpha_i+\\alpha_j)(1+\\alpha_i+\\alpha_j)/(\\alpha_i-\\alpha_j)$, in which all denominators cancel to leave integer-coefficient symmetric polynomials. Their role is to satisfy the three interpolation conditions that uniquely determine equivariant CSM classes: after restriction to the stabilizer of $\\Sigma_{n,r}$ they equal $c(T\\Sigma_{n,r})e(N\\Sigma_{n,r})$, their restrictions to all other orbits are divisible by the corresponding total Chern class, and their degrees there are strictly smaller than the Euler-class degree. The supporting machinery is the $\\Phi$-class sieve: $\\Phi$-classes are push-forwards from fibered resolutions over Grassmannians, computed by equivariant localization, and the sieve step inverts triangular matrices of binomial and Euler numbers to isolate the SSM classes.","core_discovery":"The central claim is that in the $GL_n(\\mathbb{C})$-equivariant cohomology of $\\Lambda^2\\mathbb{C}^n$ and $S^2\\mathbb{C}^n$, the CSM classes of the rank strata $\\Sigma^\\wedge_{n,r}$ and $\\Sigma^S_{n,r}$ are equal to the explicitly defined symmetric polynomials $W^\\wedge_{n,r}$ and $W^S_{n,r}$ (Theorem 5.5). Equivalently, the Segre versions are finite alternating sums of $\\Phi$-classes: $\\mathrm{ssm}(\\Sigma^\\wedge_{n,r}) = \\sum_{i=0}^{(n-r)/2} \\binom{r+2i}{r} E_{2i}\\Phi^\\wedge_{n,r+2i}$ with Euler numbers $E_{2i}$, and $\\mathrm{ssm}(\\Sigma^S_{n,r}) = \\sum_{i=0}^{n-r} (-1)^i \\binom{r+i}{r}\\Phi^S_{n,r+i}$ (and a second binomial variant). The identification holds for all $n$ and $r$, giving closed formulas for the equivariant and non-equivariant characteristic classes of both families of determinantal varieties.","pith_inferences":["If the symmetric-case identification is proved by the missing analogous argument, the same degree-counting should work with the symmetric stabilizer; a direct proof would remove the current asymmetry between the two families.","The alternating signs in Schur expansions suggest the SSM classes can be seen as differences of positive combinations; if the positivity conjecture holds, it would put symmetric and skew-symmetric degeneracy loci in the same positivity framework as other quiver and Schubert loci.","The stabilization to $n=\\infty$ suggests the formulas define universal classes in infinitely many variables; one could test whether the limit power series satisfy functional equations or specialize to known generating functions for Euler numbers or binomial transforms.","The $q$-Euler-number version in K-theory hints at a quantization of the whole sieve mechanism; a natural test is whether the $y=1$ specialization of the motivic Segre formula recovers the K-theoretic fundamental class exactly."],"forward_implications":["Substituting $\\alpha_i\\mapsto \\xi/2$ in the formulas gives the ordinary CSM classes of the projectivized loci $\\mathbb{P}\\Sigma^\\wedge_{n,r}$ and $\\mathbb{P}\\Sigma^S_{n,r}$, so the Euler characteristic of every general linear section can be read off from coefficients of $\\xi$ via the $J$-involution.","The formulas stabilize in $n$, so the limit objects $\\mathrm{ssm}(\\Sigma_{\\infty,r})$ are well-defined formal power series, and generating functions in the iterated-residue sense are obtained in a referenced thesis.","The Schur expansions of the SSM classes exhibit alternating signs, and the $\\tilde{s}_\\lambda$ expansions exhibit transpose-invariance and positivity/alternation patterns; these are stated as conjectures and supported by computed examples.","The sieve approach carries over to K-theory: the motivic Segre class $\\mathrm{mS}(\\Sigma^\\wedge_{n,r})$ is expressed with $q$-Euler numbers, providing a K-theoretic analogue of the cohomological formula.","Known enumerative data for the projectivized orbits—codimensions, degrees, and Euler characteristics—are recovered from the lowest- and top-degree terms of the formulas."],"supporting_citations":[{"why":"Provides the interpolation characterization of equivariant CSM classes and the $\\Phi$-class machinery on which both the sieve and interpolation formulas rest.","marker":"[FR2]"},{"why":"Supplies the interpolation viewpoint for equivariant CSM classes in partial flag varieties that Section 5 adapts to $\\Lambda^2\\mathbb{C}^n$ and $S^2\\mathbb{C}^n$.","marker":"[R V]"},{"why":"The thesis containing the symmetric-case proof, the Euler-obstruction formulas, generating functions, and the K-theoretic sieve formula; the paper explicitly refers to it for arguments left to the reader.","marker":"[P]"},{"why":"Classical formula for fundamental classes of skew-symmetric degeneracy loci, the base term that the CSM deformation refines.","marker":"[JLP]"},{"why":"Classical formula for fundamental classes of symmetric and skew-symmetric determinantal varieties, also refined by the CSM formulas.","marker":"[HT]"},{"why":"Establishes the equivariant CSM class and its pull-back property, used in Section 7 when passing to projectivizations.","marker":"[O1]"},{"why":"Aluffi's theorem relating the CSM class polynomial to Euler characteristics of general linear sections via the $J$-involution; the paper uses it to turn the formulas into enumerative data.","marker":"[A1]"}],"fun_headline_variants":["Explicit CSM classes for symmetric and skew degeneracy loci","Closed formulas for Chern classes of matrix rank strata","Symmetric polynomials yield Chern classes of degeneracy loci","Alternating sums describe CSM classes of symmetric loci","Equivariant CSM classes via W-functions and sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The full theorem relies on the assertion that the symmetric-matrix case follows from arguments analogous to the skew-symmetric case; if the symmetric case hides extra cancellations or different stabilizer computations, the stated symmetric formulas would need to be changed.","fun_headline_variants_meta":{"raw":{"variants":["Explicit CSM classes for symmetric and skew degeneracy loci","Closed formulas for Chern classes of matrix rank strata","Symmetric polynomials yield Chern classes of degeneracy loci","Alternating sums describe CSM classes of symmetric loci","Equivariant CSM classes via W-functions and sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000571,"raw_usage":{"total_tokens":2631,"prompt_tokens":810,"completion_tokens":1821,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":426,"completion_tokens_details":{"reasoning_tokens":1742}},"tokens_in":426,"tokens_out":1821,"duration_ms":13958,"temperature":1.0,"reasoning_tokens":1742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:19:31.409580+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\varphi_{\\Sigma^S_{n,r}}(W^S_{n,r})$ for a small case such as $(n,r)=(3,1)$ and compare with $c(T\\Sigma^S_{3,1})e(N\\Sigma^S_{3,1})$; any discrepancy would disprove Theorem 5.5. Independently, compare $W^S_{3,1}$ with the CSM class obtained from the resolution-sieve formula for $S^2\\mathbb{C}^3$.","supporting_citations":[],"review_version":1}