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Log-concavity of polynomials arising from equivariant cohomology

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Pith's one-line read This paper proves that double Richardson polynomials become covolume polynomials after a sign change, and that sign-changed equivariant classes of torus-stable subvarieties of matrix spaces are covolume.

desk verdict Real new results on log-concavity for double Schubert and Richardson polynomials, but the twisted-grading setup has a load-bearing typo that needs fixing first. read the letter →

arxiv 2411.17572 v2 pith:Q77JZKQM submitted 2024-11-26 math.AG math.ACmath.CO

Yairon Cid-Ruiz , Yupeng Li , Jacob P. Matherne This is my paper · ORCID
classification math.AGmath.ACmath.CO MSC 14M1514C1514C1713H1552B40
keywords equivariantcohomologymultidegreesRichardsonpolynomialsSchubertLorentziancovolumelog-concavityMacaulaydualgenerators
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 aims to prove that certain polynomials attached to torus-equivariant subvarieties of matrix spaces — in particular the double Richardson, Richardson, double Schubert, and Schubert polynomials of type A — are covolume polynomials after a sign change. Covolume polynomials are limits of the Chow classes of irreducible subvarieties of products of projective spaces, and they form a subfamily of dually Lorentzian polynomials. Because dually Lorentzian polynomials have M-convex support and discretely log-concave coefficients, the proof directly yields new log-concavity and support-shape statements for Schubert-related polynomials. The paper also develops Macaulay inverse systems over the integers for cohomology rings, proving that the Macaulay dual generator of the even cohomology ring of a smooth complex variety is a denormalized Lorentzian polynomial under nefness hypotheses, and extending the classical volume-polynomial description of toric cohomology to integer coefficients.

What carries the argument

The load-bearing mechanism is standardization of multigraded polynomial rings: a positive $\mathbb{N}^p$-grading is converted to a standard multigrading by replacing each variable $x_i$ of total degree $\ell_i$ with a product $y_{i,1}\cdots y_{i,\ell_i}$ of standard-graded variables. This substitution preserves multidegree polynomials, Betti numbers, and primality, so the multidegree polynomial of a prime ideal becomes the covolume polynomial of an irreducible subvariety of a product of projective spaces. A sign-flip lemma (Lemma 4.4) transfers the result from positive gradings to the 'twisted positive' gradings that arise from equivariant cohomology of matrix Schubert and Richardson varieties. The Macaulay dual generator construction over $\mathbb{Z}$ plays the analogous role for cohomology rings, replacing field-level Gorenstein duality with a relative Gorenstein statement over $\mathbb{Z}$.

What would settle it

Compute the sign-changed equivariant class polynomial of an irreducible $T$-subvariety of $\operatorname{Mat}_{m,n}$ — for instance a matrix Schubert variety — and check whether its coefficient support is M-convex; a single such class with non-M-convex support or with adjacent coefficients violating $a_n^2 \geq a_{n+e_i-e_j}a_{n-e_i+e_j}$ would refute Theorem C. Alternatively, verify directly on the action $(g,h)\cdot M = g M h^{-1}$ that the weights $e_i - e_j$ are not all in $\mathbb{N}^p$ and not all in $-\mathbb{N}^p$, which shows the literal Setup 4.3 cannot be the hypothesis under which Corollary 4.6 is proven.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem C: for the space $\operatorname{Mat}_{m,n} = \mathbb{C}^{m\times n}$ with the torus $T = (\mathbb{C}^*)^m \times (\mathbb{C}^*)^n$ acting by $(g,h)\cdot M = g M h^{-1}$, the polynomial representing the equivariant class $[X]_T$ of any irreducible $T$-subvariety $X$ becomes a covolume polynomial after the substitution $s_i \mapsto -s_i$. From this, Theorem A states that for permutations $u,w$ with $w \geq u$ in Bruhat order, the double Richardson polynomial $R_{w/u}(t,s) = S_u(t,s)S_{w_0w}(t,s')$, where $s'$ reverses the $s$ variables, has the property that $R_{w/u}(t,-s)$ is covolume. Corollary B then extracts concrete combinatorial content: these sign-changed polynomials, together with ordinary Richardson and Schubert polynomials and their truncations, have M-convex support and are discretely log-concave. In the cohomology-ring half of the paper, Theorem D establishes that for a smooth complex variety with $\mathbb{Z}$-torsion-free even cohomology, the even cohomology ring is Artinian Gorenstein over $\mathbb{Z}$, is recovered as the annihilator of a Macaulay dual generator, and that generator's normalization is Lorentzian when the chosen generators are nef first Chern classes.

Load-bearing premise

The load-bearing premise is that the torus weights can be split into a nonnegative group and a nonpositive group so that flipping signs gives a positive grading, a condition that the matrix-action weights satisfy only under the intended $\mathbb{N}^q \times (-\mathbb{N})^{p-q}$ reading rather than the literal printed Setup 4.3.

Editorial extensions

If this is right

  • The sign-changed double Richardson polynomials $R_{w/u}(t,-s)$ are dually Lorentzian, hence their coefficient supports are integer points of generalized permutohedra and are discretely log-concave.
  • Double Schubert polynomials $S_u(t,-s)$ inherit the same properties; the discrete log-concavity of double Schubert polynomials is new, while the M-convexity recovers earlier results.
  • Ordinary Schubert and Richardson polynomials, along with their truncations, have M-convex support and are discretely log-concave.
  • For smooth complex varieties with torsion-free even cohomology, the cohomology ring over $\mathbb{Z}$ is determined by a Macaulay dual generator; when the ring generators are nef first Chern classes, the normalization of that generator is Lorentzian.
  • For smooth complete toric varieties, the Macaulay dual generator is exactly the mixed-volume polynomial of the polytopes associated to nef torus-invariant divisors, giving a characteristic-free extension of the classical toric volume-polynomial description.

Reading between the lines

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

  • If Theorem C extends beyond matrix spaces, the same standardization-and-sign-flip mechanism should yield covolume polynomials for equivariant classes of torus-stable subvarieties in other representations whose weights are sign-separable, such as quiver representations with bipartite orientation.
  • The paper's truncation result suggests that the skew Schubert polynomials of Lenart and Sottile, which are normal-form representatives of Richardson classes, may be dually Lorentzian even where they do not equal truncations of Richardson polynomials; Question 5.10 is likely to have a positive answer.
  • The integral Macaulay dual generator formalism may provide a route to Lorentzian and log-concavity statements for cohomology rings with torsion by passing to a universal coefficient or flat-approximation statement, though the paper requires $\mathbb{Z}$-torsion-freeness.
  • A concrete testable extension would be to replace cohomology by equivariant K-theory and ask whether the sign-changed K-class polynomial, after normalization, is dually Lorentzian; the paper does not address K-theory.
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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

2 major / 5 minor

Summary. The paper studies torus-equivariant cohomology classes of invariant subvarieties of affine spaces and of matrix spaces. It introduces double Richardson polynomials R_{w/u}(t,s) and proves (Theorem 5.4) that the sign-changed polynomial R_{w/u}(t,-s) is a covolume polynomial, hence dually Lorentzian; this yields M-convex support and discrete log-concavity for double Richardson, Richardson, double Schubert, and Schubert polynomials (Corollary 5.8). The engine is a general statement (Theorem 4.5, Corollary 4.6) that for certain "twisted positive" torus actions on C^n, the sign-changed equivariant class of any irreducible invariant subvariety is covolume. The second half develops Macaulay inverse systems over Z for cohomology rings and proves (Theorem 6.13) that, under flatness and positivity hypotheses, the Macaulay dual generator is a denormalized Lorentzian polynomial; a toric corollary gives a characteristic-free Khovanskii–Pukhlikov description in terms of mixed volumes.

Significance. If the results hold, the paper provides a large new supply of covolume polynomials — sign-changed equivariant classes under twisted positive torus actions — and unifies several previously known log-concavity results for Schubert and double Schubert polynomials while establishing the new discrete log-concavity of double Schubert and Richardson polynomials. The characteristic-free Macaulay dual generator over Z is a useful extension of the Khovanskii–Pukhlikov theorem, and the worked examples (flag variety, Grassmannian, Hirzebruch surface) make the constructions concrete. The proofs are largely self-contained and rest on established techniques: standardization for nonstandard gradings, generic local duality over Z, and the Lorentzian volume polynomial theorem of Brändén–Huh. The paper is well structured and the computations in the examples are reproducible.

major comments (2)
  1. [Setup 4.3, Lemma 4.4, Theorem 4.5, Corollary 4.6] The twisted positive grading condition is misprinted and, read literally, excludes the main application. The text requires d_i ∈ N^p \ {0} for 1 ≤ i ≤ q and −d_i ∈ N^p \ {0} for q+1 ≤ i ≤ p, which is ill-typed (i indexes variables on the left but torus coordinates on the right) and is not satisfied by the matrix-action weights e_i − e_j ∈ Z^{m+n} used in Corollary 4.6: each such weight has a positive coordinate and a negative coordinate, so it lies in neither N^p nor −N^p. The intended condition is that every variable weight d_i lies in N^q × (−N)^{p−q}; under that reading Lemma 4.4 and Theorem 4.5 are valid and Corollary 4.6 follows. Because the printed assumption does not cover the matrix action, the proof of Theorem 5.4(ii) is not valid as written. Please correct Setup 4.3, the definition of R̃ in Lemma 4.4, and the statement of Theorem 4.5 so that the condition applies coordinate-wise to all n weights.
  2. [Theorem 6.13 and Theorem D (Introduction)] The theorem states that X is an arbitrary smooth complex algebraic variety of dimension d with R = ⊕ H^{2i}(X,Z) flat over Z, and defines ρ: H^{2d}(X,Z) → Z as the "natural degree map". For non-complete X such a map need not exist: for X = A^d_C, H^{2d}(X,Z) = 0; for X = C^*, H^2(X,Z) = 0. The statement must add that X is complete (proper); then, for connected X, H^{2d}(X,Z) ≅ Z with the usual degree map. Parts (i)–(iii) rely on Poincaré duality for R = H^*(X,Z), which also requires completeness, so the theorem as stated is false. This is load-bearing for Theorem D and Corollary 6.16. The fix is local — add the completeness hypothesis to the statements in the introduction and in Theorem 6.13 — but without it the statement overreaches.
minor comments (5)
  1. [Corollary 4.6] The phrase "irreducible T-variety" should be "irreducible T-subvariety", matching the definition in Theorem 4.5.
  2. [Setup 4.3 and Lemma 4.4] The symbol p is overloaded: it denotes the dimension of the torus in T = (C*)^q × (C*)^{p−q} and also appears as an index bound in the conditions on d_i, where the number of variables is n. Please distinguish the torus dimension from n throughout this section.
  3. [Lemma 4.4] The definition of the ring R̃ is garbled for the same reason as Setup 4.3; once the corrected twisted-positive condition is in place, R̃ should be defined on all variables by flipping the signs of the last p−q coordinates of each weight vector. The proof itself is correct after this clarification.
  4. [Corollary 6.16(i)] The symbol I is used both for the annihilator ideal {g | g·N^{−1}(V) = 0} and for the Stanley–Reisner ideal I in the presentation Z[x_1,...,x_n]/I ≅ H^*(X_Σ,Z). This reuse is confusing; please rename one of the ideals.
  5. [Proof of Lemma 4.4] The reference "[MS05, Claim 8.54]" appears to be to an unnumbered claim or to Proposition 8.54 in Miller–Sturmfels; please verify and cite the exact item.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation chain is self-contained against published theorems, with a non-circular typo caveat in Setup 4.3.

full rationale

The claimed results are derived from definitions and independent published theorems rather than from fitted inputs or self-referential definitions. Theorem 3.5 proves that multidegree polynomials of prime ideals in positive N^p-gradings are covolume by standardization (Theorem 3.4, cited to [CCRC23, Theorem 7.2] and [CCRMMn23, Proposition 4.2], both published with proofs); this is not circular because the standardization construction compares two multidegree polynomials and the cited equalities are proven statements. Lemma 4.4 flips signs in the K-polynomial to relate a twisted-positive grading to a positive one; Theorem 4.5 then combines Remark 4.1, Lemma 4.4, and Theorem 3.5. Corollary 4.6 applies this to Mat_{m,n}; the matrix Richardson variety class is computed from independent double Schubert polynomial results (Theorem 5.1, Lemma 5.2). Corollary B's log-concavity and M-convexity conclusions use the external Brändén–Huh and Ross–Süß–Wannerer/Aluffi theorems. The self-citations to [CCRMMn23], [CCRC23], and [HMMSD22] are to published independent theorems with proofs, not to an unverified premise, so they do not make the argument circular. One non-circular correctness caveat: Setup 4.3 as printed ("we require that d_i \in N^p \setminus \{0\} for all 1 \leq i \leq q and -d_i \in N^p \setminus \{0\} for all q+1 \leq i \leq p") is ill-typed and, read literally, does not cover the weights e_i - e_j of Corollary 4.6; the intended condition is evidently that every weight lies in N^q \times (-N)^{p-q}. This is a repairable misprint affecting the main application's proof, but it is not circular equivalence.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters are fitted. The central claims rest on several cited theorems (standardization, volume polynomial Lorentzian property, equivariant class formulas, generic local duality, toric presentation). These are published background results, not ad hoc assumptions. Two introduced objects (double Richardson polynomials, Macaulay dual generator over Z) are definitions with checkable geometric content.

assumptions (6)
  • domain assumption Standardization theorem: for a positively N^p-graded ring R and homogeneous ideal I, the standardized ideal J = φ(I)S in the standard multigraded ring S satisfies codim(I)=codim(J), C(R/I;t)=C(S/J;t), and J is prime if I is prime and contains no variable.
    Invoked as Theorem 3.4, cited from [CCRC23, Theorem 7.2] and [CCRMMn23, Proposition 4.2]; this is the engine behind Theorem 3.5.
  • standard math Volume polynomials of projective varieties with nef line bundles are Lorentzian.
    Used in Lemma 3.2 via [BH20, Theorem 4.6] and in Theorem 6.13(iv) after Chow's lemma.
  • domain assumption The equivariant class of a matrix Schubert variety D_w is the double Schubert polynomial S_w(t,s).
    Theorem 5.1, cited from [FR03, KM05, AF24]; foundational for deriving the double Richardson polynomial as an equivariant class.
  • domain assumption Matrix Richardson varieties D_u^w are reduced, irreducible T-subvarieties of Mat_{n,n} of dimension ℓ(w)−ℓ(u).
    Used in Theorem 5.4 to apply Corollary 4.6; cited to [Ric92, Bri05, Spe23].
  • domain assumption Generic graded local duality over a Noetherian base ring A: Ext^{n-i}_S(R, S(−δ)) ≅ *Hom_A(H^i_m(R), A) for finite flat R.
    Invoked in the proof of Theorem 6.9 as [CR23, Theorem A]; underpins the Macaulay dual generator over Z.
  • domain assumption The Jurkiewicz-Danilov presentation gives H^*(X_Σ,Z) ≅ Z[x_1,...,x_n]/(I+J) and H^*(X_Σ,Z) is Z-torsion-free for smooth complete toric varieties.
    Used in Corollary 6.16; cited to [Jur80] and [Dan78].
invented entities (2)
  • Double Richardson polynomial R_{w/u}(t,s) independent evidence
    purpose: Polynomial defined as S_u(t,s) S_{w0w}(t,s') representing the equivariant class of the matrix Richardson variety; central to Theorem A.
    Its geometric interpretation is checkable: it must equal the equivariant fundamental class of D_u^w, computable in examples.
  • Macaulay dual generator over Z independent evidence
    purpose: The inverse polynomial G_R whose S-annihilator is the presenting ideal I of the integer cohomology ring R.
    It is explicitly constructed from the degree map and computed in several examples (Fℓ3, Gr(2,4), Hirzebruch surface); the annihilator property is Theorem 6.13(iii).

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Pith. "Pith review of Log-concavity of polynomials arising from equivariant cohomology." pith.science (2026). https://pith.science/paper/Q77JZKQM

@misc{pith2026241117572,
  author       = {Pith},
  title        = {Pith review of: Log-concavity of polynomials arising from equivariant cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q77JZKQM}},
  note         = {Machine review of arXiv:2411.17572}
}
abstract

We study the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we prove that the polynomials representing these classes (up to suitably changing signs) are covolume polynomials in the sense of Aluffi. We study the cohomology rings of complex varieties in terms of Macaulay inverse systems over $\mathbb{Z}$. As applications, we show that under certain conditions, the Macaulay dual generator is a denormalized Lorentzian polynomial in the sense of Br\"and\'en and Huh, and we give a characteristic-free extension (over $\mathbb{Z}$) of the result of Khovanskii and Pukhlikov describing the cohomology ring of toric varieties in terms of volume polynomials.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Supersymmetric Schur polynomials have saturated Newton polytopes

    math.CO 2025-07 reject novelty 5.0 of 10

    A proof that supersymmetric Schur polynomials have SNP is invalid because the stated hook-inequality support set is contradicted by the paper's own example and by symmetry.

Reference graph

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