Dual Chow polynomials of matroids and posets

Elena Hoster; Giovanni Caiolo; Luis Ferroni

REVIEW 1 major objections 2 minor 12 references

Dual Chow polynomials of matroids and posets

T0 review · 1 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Dual Chow functions from sign-twisted reverse kernels yield distinct invariants for matroids when skew-symmetry fails.

desk verdict The dual Chow construction via sign-twisted reverse kernels produces a distinct invariant for matroids when skew-symmetry fails, plus a general ab-index deletion formula that looks usable. read the letter →

arxiv 2605.28474 v1 pith:PD7R2XKY submitted 2026-05-27 math.CO

Giovanni Caiolo , Luis Ferroni , Elena Hoster This is my paper · ORCID

classification math.CO MSC 05B3506A07

keywords dualChowpolynomialsmatroidsposetsincidencealgebrasab-indexgamma-positivitydeletionformulascharacteristickernel

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 the dual Chow function of a kernel in the incidence algebra of a weakly ranked poset as the Chow function of the sign-twisted reverse kernel. When the kernel lacks skew-symmetry, this produces a genuinely different polynomial invariant. For the characteristic kernel on posets arising from matroids, the authors establish chain formulas, unimodality and gamma-positivity results, formulas under standard operations, and deletion formulas, while also deriving a general deletion formula for the ab-index of matroids.

What carries the argument

The sign-twisted reverse kernel, which associates to any kernel its dual Chow function in the incidence algebra.

What would settle it

A specific matroid whose dual Chow polynomial from the characteristic kernel violates the claimed unimodality or fails a stated deletion formula.

Watch

Extended reading notes

Core claim

We introduce dual Chow functions associated to kernels in incidence algebras of weakly ranked posets. The dual Chow function is the Chow function associated to the sign-twisted reverse kernel. For kernels satisfying skew-symmetry, such as the Eulerian kernel, this recovers the ordinary Chow function; when skew-symmetry fails, as with the characteristic function on matroid posets, it gives a different invariant for which we prove chain formulas, unimodality, gamma-positivity, and deletion formulas.

Load-bearing premise

Kernels exist in the incidence algebras of weakly ranked posets and the sign-twisted reverse kernel is well-defined and yields a valid Chow function.

Editorial extensions

If this is right

Chain formulas hold for dual Chow polynomials of matroids.
Dual Chow polynomials of matroids are unimodal and gamma-positive.
Formulas under standard poset operations apply to dual Chow polynomials.
Deletion formulas hold for dual Chow polynomials of matroids.
A general deletion formula for the ab-index of matroids is obtained as a byproduct.

Reading between the lines

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

The ab-index deletion formula may enable new explicit computations of extended ab-indices for additional matroid families.
Results on dual Chow functions for matroids suggest analogous constructions could apply to other classes of weakly ranked posets.
The distinction between ordinary and dual Chow functions when skew-symmetry fails may illuminate relationships among other kernel-based invariants in incidence algebras.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core contribution is the definition of dual Chow functions from sign-twisted reverse kernels in the incidence algebra of weakly ranked posets. When the kernel is not skew-symmetric, as with the characteristic function on matroid posets, this yields an invariant different from the ordinary Chow function. The paper also derives a general deletion formula for the ab-index of matroids that specializes to several existing formulas.

They work out the expected properties: chain formulas, unimodality and gamma-positivity, and behavior under deletion and other standard operations. The ab-index deletion result is presented as new and is used to obtain formulas for extended ab-indices. This part reads as concrete and potentially applicable for people computing these invariants on matroids.

The main soft spot is the reliance on the sign-twisted reverse kernel being well-defined and producing a non-degenerate Chow function for the specific posets that arise from matroids. The abstract flags the characteristic function as the running example where skew-symmetry fails, so the paper must contain an explicit check that the algebraic conditions hold without collapse; if that verification is only sketched, it would be worth tightening. No other load-bearing gaps are visible from the claims.

This is for combinatorialists already working with matroid Chow polynomials, ab-indices, or incidence-algebra methods. It is a focused incremental piece with new formulas rather than a broad reworking of the area.

I would send it to peer review. The statements are precise enough that referees can verify the kernel conditions and the deletion identities directly.

Referee Report

1 major / 2 minor

Summary. The paper introduces dual Chow functions associated to kernels in incidence algebras of weakly ranked posets, defined as the Chow function of the sign-twisted reverse kernel. For skew-symmetric kernels it recovers the ordinary Chow function, but for the characteristic function on matroid posets (where skew-symmetry fails) it yields a distinct invariant. The authors develop the theory with emphasis on matroids, proving chain formulas, unimodality and gamma-positivity, formulas under poset operations, deletion formulas for matroids, and a general deletion formula for the ab-index of matroids that yields new formulas for extended ab-indices.

Significance. If the algebraic constructions hold, the dual Chow functions supply new invariants for matroids and posets with accompanying positivity and deletion properties. The general deletion formula for ab-indices is a concrete strength that specializes to several existing formulas in the literature and may find broader use in combinatorial enumeration.

major comments (1)

[§2 (definition of dual Chow function and main example)] The central claims (distinct invariant when skew-symmetry fails, chain formulas, positivity results, deletion formulas) all rest on the sign-twisted reverse kernel for the characteristic function being well-defined and producing a valid Chow function on matroid posets. The manuscript should supply an explicit verification that the algebraic conditions in the incidence algebra are satisfied for this kernel (rather than asserting existence), as this is load-bearing for the main example.

minor comments (2)

[§1] Notation for the sign-twisted reverse kernel could be introduced with a displayed equation to improve readability when it is first used.
[§5] The statement of the general ab-index deletion formula would benefit from an explicit comparison table to the known special cases it recovers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the recommendation for major revision. We address the single major comment below by agreeing to strengthen the exposition with an explicit verification.

read point-by-point responses

Referee: [§2 (definition of dual Chow function and main example)] The central claims (distinct invariant when skew-symmetry fails, chain formulas, positivity results, deletion formulas) all rest on the sign-twisted reverse kernel for the characteristic function being well-defined and producing a valid Chow function on matroid posets. The manuscript should supply an explicit verification that the algebraic conditions in the incidence algebra are satisfied for this kernel (rather than asserting existence), as this is load-bearing for the main example.

Authors: We agree that the manuscript would benefit from an explicit check rather than relying on the general definition. In the revised version we will insert a short lemma (or dedicated paragraph) in §2 that directly verifies the two incidence-algebra axioms (multiplicativity over intervals and the required normalization) for the sign-twisted reverse of the characteristic function on the lattice of flats of a matroid. This verification uses only the standard properties of the characteristic function and the sign-twisting by rank, both of which are already introduced earlier in the section. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: definitions and derivations are self-contained from incidence algebra operations

full rationale

The paper introduces dual Chow functions via the sign-twisted reverse kernel on weakly ranked posets, using standard incidence-algebra constructions. All subsequent results (chain formulas, unimodality, gamma-positivity, deletion formulas, ab-index formulas) are derived from these definitions without reducing any claimed prediction or theorem to a fitted parameter, self-citation chain, or input-by-construction equivalence. The characteristic function example is presented as a case where skew-symmetry fails, but the algebraic well-definedness is taken as an assumption rather than derived circularly from the results themselves. No load-bearing self-citations or ansatzes smuggled via prior work are present in the derivation chain.

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

No free parameters or invented entities; the work rests on standard definitions of incidence algebras and weakly ranked posets.

assumptions (2)

standard math Weakly ranked posets admit well-defined incidence algebras in which kernels can be defined.
Standard background assumption in poset combinatorics invoked to define the Chow and dual Chow functions.
domain assumption The sign-twisted reverse of a kernel again defines a valid Chow function.
Central modeling choice that enables the dual construction.

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Cite this review

Pith. "Pith review of Dual Chow polynomials of matroids and posets." pith.science (2026). https://pith.science/paper/PD7R2XKY

@misc{pith2026260528474,
  author       = {Pith},
  title        = {Pith review of: Dual Chow polynomials of matroids and posets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PD7R2XKY}},
  note         = {Machine review of arXiv:2605.28474}
}

read the original abstract

We introduce and study dual Chow functions associated to kernels in incidence algebras of weakly ranked posets. Given a kernel, its dual Chow function is defined as the Chow function associated to the sign-twisted reverse kernel. For kernels satisfying a natural skew-symmetry condition, such as the Eulerian kernel of an Eulerian poset or the kernel given by R-polynomials on Bruhat intervals, this construction recovers the ordinary Chow function. In contrast, when this skew-symmetry fails, the dual Chow function gives a genuinely different invariant. The main example considered in this paper is the dual Chow function associated to the characteristic function. We develop the basic theory of these dual Chow functions, with particular emphasis on posets arising from matroids. We prove chain formulas, unimodality and gamma-positivity results, formulas under standard poset operations, and deletion formulas for matroids. Along the way, we also obtain a general deletion formula for the ab-index of matroids, which leads to new formulas for extended ab-indices and, in turn, specializes to several deletion formulas appearing in the literature.

Figures

Figures reproduced from arXiv: 2605.28474 by the authors.

**Figure 1.** A poset whose dual Chow polynomial has both plus and minus signs The preceding example hints at the requirement we need to impose in order to guarantee a “sign-definiteness” for the coefficients in H∗ 𝑃 (𝑥). By Theorem 2.5(i), we know that the following holds for any weakly ranked poset (𝑃, 𝜌): [𝑥 𝜌𝑠𝑡 −1 ]H ∗ 𝑠𝑡(𝑥) = (−1) 𝜌𝑠𝑡 [𝑥 𝜌𝑠𝑡 ] 𝜒 rev 𝑠𝑡 (𝑥) = (−1) 𝜌𝑠𝑡 𝜇𝑠𝑡 . The Mobius function ¨ alternates in sign if it satis… view at source ↗

**Figure 2.** The lattice of flats of the uniform matroid U3,4. Example 4.5 Let Π𝑛 denote the partition lattice of the set [𝑛 + 1], ordered by refinement. We regard Π𝑛 as a graded lattice of rank 𝑛, with minimum element the partition into singletons and with maximum element the trivial partition. The first values of the dual Chow polynomial are as follows: H ∗ Π𝑛 (𝑥) =    1 𝑛 = 1, 2𝑥 + 2 𝑛 = 2, 6𝑥 2 + 18𝑥… view at source ↗

**Figure 3.** Three extensions of ab-index variations. 4.4. Gamma-positivity. For a subset 𝑆 ⊆ [𝑟 − 1], let 𝑆 𝑐 = [𝑟 − 1] \ 𝑆. We shall say that 𝑆 is stable if it contains no two consecutive integers. Theorem 4.9 Let 𝑃 be a bounded weakly ranked poset of rank 𝑟. The polynomials H ∗ 𝑃 (𝑥) and 𝐹 ∗ 𝑃 (𝑥) can be written as follows. H ∗ 𝑃 (𝑥) = ∑︁ 𝑆⊆ [𝑟−1] 𝑟−1∉𝑆 𝑆 stable 𝛽𝑃(𝑆 𝑐 ) 𝑥 |𝑆| (1 + 𝑥) 𝑟−1−2|𝑆| , and 𝐹 ∗ 𝑃 (𝑥) = ∑︁ 𝑆⊆ [𝑟−1] 𝑆 … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A rank 4 poset satisfying the Mobius sign condition, but whose dual ¨ Chow polynomial is not 𝛾-positive. Remark 4.12 (Dual Chow polynomials of partitions lattices of types A and B) Using Theorem 4.9 and following [Stu25], we can the express dual Chow polynomials of th…

**Figure 5.** Figure 5: An EL-shellable (thus Cohen–Macaulay) poset whose dual Chow polynomial is not real-rooted [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]

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Reference graph

Works this paper leans on

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Athanasiadis, Theo Douvropoulos, and Katerina Kalampogia-Evangelinou,Two classes of posets with real-rooted chain polynomials, Electron

[ADKE24] Christos A. Athanasiadis, Theo Douvropoulos, and Katerina Kalampogia-Evangelinou,Two classes of posets with real-rooted chain polynomials, Electron. J. Combin.31(2024), no. 4, Paper No. 4.16,

2024

[2] [2]

[AHK18] Karim Adiprasito, June Huh, and Eric Katz,Hodge theory for combinatorial geometries, Ann. of Math. (2)188(2018), no. 2, 381–452. [AKE23] Christos A. Athanasiadis and Katerina Kalampogia-Evangelinou,Chain enumeration, partition lattices and polynomials with only real roots, Comb. Theory3(2023), no. 1, Paper No. 12,

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[BDF22] Christin Bibby, Graham Denham, and Eva Maria Feichtner,A Leray model for the Orlik-Solomon algebra, Int. Math. Res. Not. IMRN (2022), no. 24, 19105–19174. [BFMN25] Tom Braden, Luis Ferroni, Jacob P. Matherne, and Nutan Nepal,Inverse Kazhdan-Lusztig polynomials of matroids under deletion, arXiv e-prints (2025), arXiv:2510.01086. [BHM+22a] Tom Brade...

work page arXiv 2022

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[Bre99] Francesco Brenti,Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math.148 (1999), no. 1, 44–74. [BST23] Andrew Berget, Hunter Spink, and Dennis Tseng,Log-concavity of matroidℎ-vectors and mixed Eulerian numbers, Duke Math. J.172(2023), no. 18, 3475–3520. [BV20] Tom Braden and Artem Vysogorets,Kazhdan-Lusztig polynomials of m...

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[BV26] Petter Br ¨and´en and Lorenzo Vecchi,Chow polynomials of uniform matroids are real-rooted, Comb

[BV25] Petter Br ¨and´en and Lorenzo Vecchi,Chow polynomials of totally nonnegative matrices and posets, arXiv e-prints (2025), arXiv:2509.17852. [BV26] Petter Br ¨and´en and Lorenzo Vecchi,Chow polynomials of uniform matroids are real-rooted, Comb. Theory6(2026), no. 1, Paper No. 15,

work page arXiv 2025

[6] [6]

[DBMS25] Galen Dorpalen-Barry, Joshua Maglione, and Christian Stump,The Poincar ´e-extendedab-index, J

[CFL25] Basile Coron, Luis Ferroni, and Shiyue Li,Chow polynomials of rank-uniform labeled posets, arXiv e-prints (2025), arXiv:2511.13819. [DBMS25] Galen Dorpalen-Barry, Joshua Maglione, and Christian Stump,The Poincar ´e-extendedab-index, J. Lond. Math. Soc. (2)111(2025), no. 1, Paper No. e70054, 33, With an appendix by Ricky Ini Liu. [DHM+26] Sebastian...

work page arXiv 2025

[7] [7]

Math.299(2016), 36–70

[EPW16] Ben Elias, Nicholas Proudfoot, and Max Wakefield,The Kazhdan-Lusztig polynomial of a matroid, Adv. Math.299(2016), 36–70. [ER95] Richard Ehrenborg and Margaret A. Readdy,Sheffer posets and𝑟-signed permutations, Ann. Sci. Math. Qu´ebec19(1995), no. 2, 173–196. [FMSV24] Luis Ferroni, Jacob P. Matherne, Matthew Stevens, and Lorenzo Vecchi,Hilbert-Poi...

2016

[8] [8]

[FS24] Luis Ferroni and Benjamin Schr ¨oter,Valuative invariants for large classes of matroids, J. Lond. Math. Soc. (2)110(2024), no. 3, Paper No. e12984. [FY04] Eva Maria Feichtner and Sergey Yuzvinsky,Chow rings of toric varieties defined by atomic lattices, Invent. Math.155(2004), no. 3, 515–536. [Gal05] Swiatoslaw R. Gal,Real root conjecture fails for...

work page arXiv 2024