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arxiv: 2606.11005 · v1 · pith:CNEDDBU3new · submitted 2026-06-09 · 🧮 math.CO

On zero-sum polytopes: reciprocity, rigidity, and cyclic sieving

Dongchun Han , Xuan Wang , Hanbin Zhang , Shiwen Zhang This is my paper

Pith reviewed 2026-06-27 12:38 UTC · model grok-4.3

classification 🧮 math.CO
keywords zero-sum sequenceszero-sum polytopesEhrhart theorycombinatorial reciprocitycyclic sievingabelian groupslattice pointspolytope faces
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The pith

Zero-sum polytopes are rigid: matching total lattice counts at dilations forces equinumerous open-face strata in each dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a rational polytope P_G of dimension n-1 for a finite abelian group G of order n so that the lattice points of its m-dilate exactly count the zero-sum sequences of length m over G. This turns the enumeration problem into one in rational Ehrhart theory and produces a reciprocity theorem in which negative evaluations of the counting quasipolynomial recover the zero-sum sequences that have full support. The central theorem asserts a face-stratified rigidity: if any two such polytopes agree on the total number of lattice points at certain dilations, then the numbers of lattice points lying in the open faces of each dimension must also agree. The same construction yields equivariant generating functions under the natural Aut(G) action and cyclic sieving for associated cyclic actions.

Core claim

Whenever two zero-sum polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. This rigidity follows from the defining bijection between lattice points of m P_G and zero-sum sequences of length m, together with the resulting quasipolynomial and its face decomposition.

What carries the argument

The zero-sum polytope P_G, whose m-dilates satisfy |m P_G ∩ Z^n| = |M(G,m)| and whose open faces stratify the support conditions on sequences.

If this is right

  • Enumeration of zero-sum sequences is realized as a counting problem in rational Ehrhart theory.
  • Negative values of the associated quasipolynomial count zero-sum sequences of full support via reciprocity.
  • The Aut(G) action produces equivariant generating functions and reciprocity formulas.
  • Natural cyclic actions on the polytopes satisfy cyclic sieving.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same rigidity may hold for other families of polytopes whose lattice points encode combinatorial objects with a natural support stratification.
  • The face-wise equinumerosity could be used to transfer enumeration results between different groups G and G' when only total counts are known.
  • Cyclic sieving phenomena obtained here might extend to other finite-group actions on zero-sum sequences beyond the automorphism group.

Load-bearing premise

The m-dilates of P_G have lattice points in exact bijection with the zero-sum sequences of length m.

What would settle it

Two distinct zero-sum polytopes that agree on total lattice-point counts for all sufficiently large m yet differ in the count of lattice points inside some open face of fixed dimension.

read the original abstract

Let $G$ be a finite abelian group of order $n$, and let $\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \[ |\mathsf M(G,m)|=|m\mathcal P_G\cap \mathbb Z^n|. \] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full support. Our main results establish a face-stratified rigidity for zero-sum polytopes: whenever two such polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. Moreover, we study the natural $\operatorname{Aut}(G)$-action on $\mathcal P_G$, derive equivariant generating functions and reciprocity formulas, and obtain cyclic sieving phenomena for natural cyclic actions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces the zero-sum polytope P_G associated to a finite abelian group G of order n, such that the number of zero-sum sequences of length m equals the lattice-point count in the m-dilate of P_G. It derives a reciprocity theorem via Ehrhart theory identifying negative evaluations with full-support zero-sum sequences, proves a face-stratified rigidity result (equal total counts at certain dilates imply equinumerous open-face strata by dimension), and obtains Aut(G)-equivariant generating functions together with cyclic sieving phenomena for natural cyclic actions.

Significance. If the central claims hold, the work supplies a new geometric model for zero-sum sequence enumeration that directly imports tools from rational Ehrhart theory, yielding reciprocity and a rigidity theorem that equates global and stratified counts. The Aut(G)-equivariant and cyclic-sieving results further connect the construction to representation theory and combinatorial enumeration under group actions. These links are potentially useful for explicit computations and symmetry detection in zero-sum problems.

minor comments (3)
  1. [Abstract] The abstract states that P_G is a rational polytope of dimension n-1 whose m-dilates realize the bijection with M(G,m), but does not indicate the ambient space or the linear dependence that reduces the dimension; a single sentence clarifying the embedding (e.g., the hyperplane sum of coordinates equals a fixed value) would improve readability for readers outside Ehrhart theory.
  2. [Introduction / Main Theorem] The rigidity statement is phrased in terms of “specific dilations”; the precise arithmetic progression or set of m for which the implication holds should be stated explicitly in the introduction or the statement of the main theorem to allow immediate verification of the hypothesis.
  3. [Section on face-stratified rigidity] Notation for the open-face strata (dimension-wise) is introduced without a displayed formula relating the strata to the faces of P_G; adding a short displayed equation or a reference to the standard face lattice of a polytope would clarify the stratification used in the rigidity result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the zero-sum polytope construction, and recommendation of minor revision. The report correctly identifies the use of rational Ehrhart theory for reciprocity, the face-stratified rigidity theorem, and the Aut(G)-equivariant cyclic sieving results. No specific major comments appear in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via definition and standard Ehrhart theory

full rationale

The paper defines the zero-sum polytope P_G explicitly so that the lattice-point equality |M(G,m)| = |m P_G ∩ Z^n| holds by construction, then invokes the standard (non-self-referential) Ehrhart reciprocity theorem for rational polytopes to obtain the negative-evaluation reciprocity for full-support sequences. The central rigidity theorem—that equal total counts at specific dilations imply equinumerous open-face strata—is a derived combinatorial statement, not a renaming or fitted prediction. No load-bearing self-citation, uniqueness theorem imported from the authors, or ansatz smuggled via prior work appears; the Aut(G)-equivariant generating functions and cyclic sieving statements follow directly from the construction and group action without reducing to the input counts by definition. The chain is therefore independent of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The construction rests on the standard definition of finite abelian groups and the new polytope whose lattice points are stipulated to encode zero-sum sequences.

axioms (1)
  • domain assumption G is a finite abelian group of order n
    Stated at the opening of the abstract; used to define M(G,m) and P_G.
invented entities (1)
  • zero-sum polytope P_G no independent evidence
    purpose: Rational polytope of dimension n-1 whose m-dilates count zero-sum sequences via lattice points
    Newly introduced object whose definition is the central modeling step.

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Reference graph

Works this paper leans on

59 extracted references

  1. [1]

    Alexandersson, S

    P. Alexandersson, S. Linusson, S. Potka and J. Uhlin, Refined Catalan and Narayana cyclic sieving,Comb. Theory1(2021), Paper No. 7, 53 pp

    2021

  2. [2]

    Almkvist and R

    G. Almkvist and R. Fossum, Decomposition of exterior and symmetric powers of indecomposableZ/pZ- modules in characteristicp and relations to invariants, in:Seminaire d’Algebre P. Dubreil, Paris, 1976–1977, Lecture Notes in Math., vol. 641, Springer, Berlin, 1978, pp. 1–111

    1976

  3. [3]

    Ardila, M

    F. Ardila, M. Supina and A. R. Vindas-Melendez, The equivariant Ehrhart theory of the permutahedron, Proc. Amer. Math. Soc.148(2020), no. 12, 5091–5107

    2020

  4. [4]

    Beck, A closer look at the number of lattice points in rational simplices,Electron

    M. Beck, A closer look at the number of lattice points in rational simplices,Electron. J. Combin.6(1999), no. 1, R37

    1999

  5. [5]

    Beck, Multidimensional Ehrhart reciprocity,J

    M. Beck, Multidimensional Ehrhart reciprocity,J. Combin. Theory Ser. A97(2002), no. 1, 187–194

    2002

  6. [6]

    M. Beck, S. Elia and S. Rehberg, Rational Ehrhart theory,Integers23(2023), Paper No. A60, 31 pp

    2023

  7. [7]

    Beck and S

    M. Beck and S. Robins,Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra, Undergraduate Texts in Mathematics, Springer, New York, 2015

    2015

  8. [8]

    Beck and R

    M. Beck and R. Sanyal,Combinatorial Reciprocity Theorems, Graduate Studies in Mathematics, vol. 195, American Mathematical Society, Providence, RI, 2018

    2018

  9. [9]

    Beck and T

    M. Beck and T. Zaslavsky, Inside-out polytopes,Adv. Math.205(2006), no. 1, 134–162

    2006

  10. [10]

    Bergeron,Algebraic Combinatorics and Coinvariant Spaces, A K Peters/CRC Press, 2009

    F. Bergeron,Algebraic Combinatorics and Coinvariant Spaces, A K Peters/CRC Press, 2009

    2009

  11. [11]

    Borger, A

    C. Borger, A. Kretschmer and B. Nill, Thin polytopes: lattice polytopes with vanishing localh∗-polynomial, Int. Math. Res. Not. IMRN2024, no. 7, 5619–5657

  12. [12]

    Clarke and M

    O. Clarke and M. Kolbl, Equivariant Ehrhart theory of hypersimplices,Forum Math. Sigma13(2025), e178, 29 pp

    2025

  13. [13]

    Cziszter, M

    K. Cziszter, M. Domokos and A. Geroldinger, The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics, in:Multiplicative Ideal Theory and Factorization Theory, Springer Proceedings in Mathematics and Statistics, vol. 170, Springer, Cham, 2016, pp. 43–95

    2016

  14. [14]

    D’Ali, M

    A. D’Ali, M. Juhnke-Kubitzke and M. Koch, On a generalization of symmetric edge polytopes to regular matroids,Int. Math. Res. Not. IMRN2024, no. 14, 10844–10864

  15. [15]

    D’Ali and E

    A. D’Ali and E. Delucchi, Equivariant Hilbert and Ehrhart series under translative group actions,J. Lond. Math. Soc.(2)112(2025), e70341

    2025

  16. [16]

    de Vries and M

    T. de Vries and M. Yoshinaga, Symmetry and the Ehrhart quasi-polynomial of rational polytopes,Int. Math. Res. Not. IMRN2022, 7513–7542

  17. [17]

    Ehrhart, Sur les polyedres rationnels homothetiques an dimensions,C

    E. Ehrhart, Sur les polyedres rationnels homothetiques an dimensions,C. R. Acad. Sci. Paris254(1962), 616–618

    1962

  18. [18]

    A. G. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups,Indag. Math.9(1998), 233–238

    1998

  19. [19]

    A. G. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of necklaces and Hermite reciprocity,J. Algebraic Combin.10(1999), 173–188

    1999

  20. [20]

    S. Elia, D. Kim and M. Supina, Techniques in equivariant Ehrhart theory,Ann. Comb.28(2024), 819–870

    2024

  21. [21]

    Fredman, A symmetry relationship for a class of partitions,J

    M. Fredman, A symmetry relationship for a class of partitions,J. Combin. Theory Ser. A18(1975), 199–202

    1975

  22. [22]

    S. Fu, D. Tang, B. Han and J. Zeng,(q, t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon,Adv. in Appl. Math.106(2019), 57–95

    2019

  23. [23]

    A. M. Garsia, M. Haiman, A remarkableq, t-Catalan sequence andq-Lagrange inversion,J. Algebraic Combin. 5(1996), no. 3, 191–244

    1996

  24. [24]

    Gao and A

    W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey,Expo. Math.24(2006), 337–369

    2006

  25. [25]

    Geroldinger, Sets of lengths,Amer

    A. Geroldinger, Sets of lengths,Amer. Math. Monthly123(2016), no. 10, 960–988. ZERO-SUM POLYTOPES 31

    2016

  26. [26]

    Geroldinger and F

    A. Geroldinger and F. Halter-Koch,Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006

    2006

  27. [27]

    Geroldinger and J

    A. Geroldinger and J. Oh, On the isomorphism problem for monoids of product-one sequences,Bull. Lond. Math. Soc.57(2025), no. 5, 1482–1495

    2025

  28. [28]

    Geroldinger, D

    A. Geroldinger, D. Grynkiewicz and Q. Zhong,Combinatorial Factorization Theory, Mathematical Surveys and Monographs, to appear, American Mathematical Society, Providence, RI, 2026

    2026

  29. [29]

    Geroldinger and Q

    A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids,Semigroup Forum100(2020), no. 1, 22–51

    2020

  30. [30]

    Gustafsson and L

    N. Gustafsson and L. Solus, Derangements, Ehrhart theory, and localh-polynomials,Adv. Math.369(2020), 107169, 35 pp

    2020

  31. [31]

    Haglund,The q, t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series, American Mathematical Society, 2007

    J. Haglund,The q, t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series, American Mathematical Society, 2007

    2007

  32. [32]

    Haiman, Conjectures on the quotient ring by diagonal invariants,J

    M. Haiman, Conjectures on the quotient ring by diagonal invariants,J. Algebraic Combin.3(1994), 17–76

    1994

  33. [33]

    Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture,J

    M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture,J. Amer. Math. Soc.14 (2001), no. 4, 941–1006

    2001

  34. [34]

    Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane,Invent

    M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane,Invent. Math.149(2002), 371–407

    2002

  35. [35]

    D. C. Han and H. B. Zhang, Erdos-Ginzburg-Ziv theorem and Noether number forCmn⊕Cmn,J. Number Theory198(2019), 159–175

    2019

  36. [36]

    D. C. Han and H. B. Zhang, A reciprocity on finite abelian groups involving zero-sum sequences,SIAM J. Discrete Math.35(2021), no. 2, 1077–1095

    2021

  37. [37]

    Hibi,Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, 1992

    T. Hibi,Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, 1992

    1992

  38. [38]

    J. C. Lagarias and G. M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice,Canad. J. Math.43(1991), no. 5, 1022–1035

    1991

  39. [39]

    M. S. Li and H. B. Zhang, The group permanent determines the finite abelian group,Electron. J. Combin. 31(2024), no. 4, Paper No. 4.44, 24 pp

    2024

  40. [40]

    I. G. Macdonald, Polynomials associated with finite cell-complexes,J. London Math. Soc.(2)4(1971), 181–192

    1971

  41. [41]

    Maxim and J

    L. Maxim and J. Schurmann, Weighted Ehrhart theory via equivariant toric geometry,Adv. Math.488 (2026), Paper No. 110771, 55 pp

    2026

  42. [42]

    T. B. McAllister and K. M. Woods, The minimum period of the Ehrhart quasi-polynomial of a rational polytope,J. Combin. Theory Ser. A109(2005), 345–352

    2005

  43. [43]

    Molien, Uber die Invarianten der linearen Substitutionsgruppe,Sitzungsber

    T. Molien, Uber die Invarianten der linearen Substitutionsgruppe,Sitzungsber. Konig. Preuss. Akad. Wiss. (1897), 1152–1156

  44. [44]

    Muratovic-Ribic and Q

    A. Muratovic-Ribic and Q. Wang, The multisubset sum problem for finite abelian groups,Ars Math. Contemp. 8(2015), no. 2, 417–423

    2015

  45. [45]

    Oh and B

    J. Oh and B. Rhoades, Cyclic sieving and orbit harmonics,Math. Z.300(2022), no. 1, 639–660

    2022

  46. [46]

    D. I. Panyushev, Fredman’s reciprocity, invariants of abelian groups, and the permanent of the Cayley table, J. Algebraic Combin.33(2011), 111–125

    2011

  47. [47]

    Reiner, D

    V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon,J. Combin. Theory Ser. A108(2004), no. 1, 17–50

    2004

  48. [48]

    Reiner, D

    V. Reiner, D. Stanton and D. White, What is ... cyclic sieving?,Notices Amer. Math. Soc.61(2014), no. 2, 169–171

    2014

  49. [49]

    Sagan, The cyclic sieving phenomenon: a survey, in:Surveys in Combinatorics 2011, London Math

    B. Sagan, The cyclic sieving phenomenon: a survey, in:Surveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser., vol. 392, Cambridge Univ. Press, 2011

    2011

  50. [50]

    R. P. Stanley, Combinatorial reciprocity theorems,Adv. Math.14(1974), no. 2, 194–253

    1974

  51. [51]

    R. P. Stanley, Decompositions of rational convex polytopes,Ann. Discrete Math.6(1980), 333–342

    1980

  52. [52]

    Stapledon, Weighted Ehrhart theory and orbifold cohomology,Adv

    A. Stapledon, Weighted Ehrhart theory and orbifold cohomology,Adv. Math.219(2008), 63–88

    2008

  53. [53]

    Stapledon, Equivariant Ehrhart theory,Adv

    A. Stapledon, Equivariant Ehrhart theory,Adv. Math.226(2011), no. 4, 3622–3654

    2011

  54. [54]

    Stapledon, Equivariant Ehrhart theory, commutative algebra and invariant triangulations of polytopes, arXiv:2311.17273

    A. Stapledon, Equivariant Ehrhart theory, commutative algebra and invariant triangulations of polytopes, arXiv:2311.17273

    arXiv

  55. [55]

    J. R. Stembridge, Some hidden relations involving the ten symmetry classes of plane partitions,J. Combin. Theory Ser. A68(1994), no. 2, 372–409

    1994

  56. [56]

    J. R. Stembridge, On minuscule representations, plane partitions and involutions in complex Lie groups, Duke Math. J.73(1994), 469–490

    1994

  57. [57]

    J. R. Stembridge, Canonical bases and self-evacuating tableaux,Duke Math. J.82(1996), no. 3, 585–606. 32 DONGCHUN HAN, XUAN W ANG, HANBIN ZHANG, AND SHIWEN ZHANG

    1996

  58. [58]

    X. Wang, H. Zhang and S. Zhang, On immanants of the Cayley table of finite abelian groups,Bull. Braz. Math. Soc. (N.S.)57(2026), no. 1, Paper No. 5

    2026

  59. [59]

    Zeng and S

    C. Zeng and S. Zhang, Cyclic sieving and dihedral sieving on noncrossing(1,2)-configurations,Discrete Math. 348(2025), no. 2, Paper No. 114262. Department of Mathematics, Southwest Jiaotong University, Chengdu 610000, P.R. China E-mail address:handongchun@swjtu.edu.cn School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, P.R. C...

    2025