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REVIEW 3 major objections 5 minor 31 references

Characteristic classes of symmetric and skew-symmetric degeneracy loci

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Pith's one-line read 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.

desk verdict 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FR2 interpolation framework with a genuine, load-bearing gap: the symmetric half of the main theorem is deferred to an unpublished thesis. read the letter →

arxiv 1908.07373 v1 pith:PRUZPKYN submitted 2019-08-20 math.AG

Sutipoj Promtapan , Richard Rimanyi This is my paper · ORCID
classification math.AG MSC 14C1714M1214L30
keywords Chern-Schwartz-MacPhersonclassesSegre-Schwartz-Macdegeneracylocisymmetricmatricesskew-symmetricequivariantcohomologyinterpolationmotivicChern
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

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.

What carries the argument

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.

What would settle it

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 $S^2\mathbb{C}^3$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§5.2, Theorem 5.5] 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.
  2. [§4.3, Theorem 4.7] 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].
  3. [§5.2 / Assumption 3.3] 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.
minor comments (5)
  1. [§4.1, proof of Proposition 4.1] There is a typo, `bunddle` for `bundle`, in the first sentence of the proof.
  2. [§4.3, Theorem 4.7] 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.
  3. [§5.2, degree estimate] 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.
  4. [§5.2, r=0 verification] There is a typo, `obtian` for `obtain`, in the computation of φ_{Σ^∧_{n,0}}(W^∧_{n,0}).
  5. [§7.3] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Symmetric half of the central theorem is deferred to the first author's unpublished thesis; the skew-symmetric derivation is self-contained.

  1. self citation load bearing [Theorem 5.5 proof, final sentence (Section 5.2); the same pattern appears before Theorem 4.7]
    "The proof of the second statement, the case of S 2Cn, is analogous, we leave it to the reader (or see [P])."

    The second sentence of Theorem 5.5, csm(Σ^S_{n,r}) = W^S_{n,r}, is the full symmetric half of the paper's central claim. Its proof is not given in the manuscript: the text only asserts analogy and points to [P], the first author's doctoral thesis, which is not independently established in the paper. The symmetric case would require separate verification of the three interpolation conditions of Theorem 3.4 for a different stabilizer (O(n−r)×GL_r), a different normal representation (S^2 of the kernel), unrestricted orbit parity, and different degree bounds; none of these checks appear in the text. Thus the symmetric family's claimed derivation reduces, for the reader of this paper, to a load-bearing self-citation rather than to a demonstrated computation.

full rationale

The leading skew-symmetric derivation is self-contained: the W^ functions are defined explicitly and then verified against the three interpolation conditions of Theorem 3.4, with the stabilizer, tangent and normal Chern classes, and degree estimates computed in the text. Theorem 3.4 is taken from the published external source [FR2]; although it shares an author, it is not re-derived from the target formula and is independent support, not circular input. No fitted parameters are introduced at any point. The sieve formula for the skew-symmetric case, Theorem 4.5, is a genuine inversion of the pushforward relation in Proposition 4.2 using the Euler-number matrix identity of Proposition 4.4. The circularity concern is confined to the symmetric half: csm(Σ^S_{n,r}) = W^S_{n,r} is asserted, with the proof deferred to the first author's thesis [P] and to an unstated analogy. This is a load-bearing self-citation for half of the central theorem, not a derivation equivalent to its input by construction. A score of 4 reflects the fact that the skew-symmetric content and the interpolation verification are independent and substantive, while the symmetric half is not established within the submitted text.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

No empirical or fitted constants appear. The central developments rest on published theorems in equivariant CSM theory, on the interpolation characterization, and on an analogy argument for symmetric matrices. The explicit W-functions are new constructions and are not independently attested outside this work.

assumptions (7)
  • standard math Existence, uniqueness, and stated properties of equivariant CSM classes (MacPherson, Ohmoto).
    Used throughout as the definitional framework for CSM and SSM classes; cited from [M, O1, O2, O3].
  • standard math Interpolation characterization of CSM classes, Theorem 3.4 from [FR2].
    The paper verifies the three interpolation conditions for W-functions; this theorem is the bridge from explicit functions to CSM classes.
  • domain assumption Assumption 3.3 holds for the representations Lambda^2 C^n and S^2 C^n: finitely many orbits, orbits are cones, and Euler class ep(N_Sigma) is nonzero.
    Verified in the proof of Theorem 5.5 for the skew-symmetric case and asserted to hold analogously for the symmetric case.
  • standard math Equivariant localization on Grassmannians computes the Phi-class integrals.
    Used in Proposition 4.1 to derive the explicit formula (3) for Phi^_n,r.
  • standard math Pull-back property (iv) of SSM classes under transverse maps.
    Used for stabilization in Section 6 and for passing from vector spaces to projective spaces in Theorem 7.1; cited from [O1, O3].
  • standard math Aluffi's theorem on the J involution relating CSM classes to Euler characteristics of linear sections.
    Applied in Section 7.3 to convert CSM classes into Euler characteristic polynomials.
  • ad hoc to paper The symmetric-case Sieve formula and W-function formulas hold by analogy with the skew-symmetric case.
    The text states that details are left to the reader or to [P], so the symmetric half of the central claim is assumed rather than demonstrated in the paper.
invented entities (1)
  • W-functions W^_n,r and W^S_n,r
    purpose: Explicit rational functions proposed as representatives of CSM classes for the skew-symmetric and symmetric orbit closures.
    The functions are explicitly defined in Section 5.1 and verified against the interpolation conditions, but they are new to this paper and have no independent attestation outside the text.

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Pith. "Pith review of Characteristic classes of symmetric and skew-symmetric degeneracy loci." pith.science (2026). https://pith.science/paper/PRUZPKYN

@misc{pith2026190807373,
  author       = {Pith},
  title        = {Pith review of: Characteristic classes of symmetric and skew-symmetric degeneracy loci},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRUZPKYN}},
  note         = {Machine review of arXiv:1908.07373}
}
read the original abstract

We give two formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric degeneracy loci. We apply them in enumerative geometry, explore their algebraic combinatorics, and discuss K theory generalizations.

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