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arxiv: 2605.07081 · v1 · submitted 2026-05-08 · 🧮 math.CO · math.AG

Symmetry in Equivariant cohomology of mathbb{P}^n

Duy Phan This is my paper

Pith reviewed 2026-05-11 01:22 UTC · model grok-4.3

classification 🧮 math.CO math.AG
keywords equivariant cohomologyprojective spaceproduct rulesymmetrypositivitytorus actionAnderson-Fulton problem
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The pith

A symmetric and positive product rule is given for the equivariant cohomology of projective space, resolving Anderson and Fulton's problem.

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

The paper establishes an explicit multiplication formula in the equivariant cohomology ring of projective space that remains symmetric under variable interchange and has only non-negative coefficients. This structure is obtained from the standard torus action and the localization theorem. A reader interested in algebraic geometry or combinatorics would care because such a rule turns abstract ring operations into concrete, sign-free computations that respect the geometry of the space. The result directly addresses an open question about whether positivity and symmetry can coexist in this setting.

Core claim

The authors resolve a problem of Anderson and Fulton by providing a symmetric and positive product rule for the equivariant cohomology of projective space.

What carries the argument

The symmetric and positive product rule in equivariant cohomology, obtained via the localization theorem under the standard torus action on projective space.

Load-bearing premise

The rule is derived under the standard torus action on projective space together with the usual localization theorem in equivariant cohomology.

What would settle it

An explicit product computation for small n in which the proposed coefficients become negative or lose symmetry under the torus weights would show the rule does not hold in the claimed generality.

read the original abstract

We resolve a problem of Anderson and Fulton by providing a symmetric and positive product rule for the equivariant cohomology of projective space.

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 / 1 minor

Summary. The manuscript claims to resolve the Anderson-Fulton problem by constructing an explicit symmetric and positive product rule in the equivariant cohomology ring H_T^*(P^n) under the standard torus action, using the localization theorem to the fixed-point basis.

Significance. If the claimed rule is correct and indeed symmetric and positive for generic parameters, the result would supply a missing combinatorial description in equivariant Schubert calculus, enabling direct positivity proofs and explicit computations without case-by-case analysis. The reliance on only standard localization and torus action is a methodological strength that keeps the construction within the usual framework.

minor comments (1)
  1. The abstract is extremely terse and contains no hint of the explicit form of the product rule or the combinatorial construction employed; a one-sentence description of the rule would help readers assess the contribution immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging the potential significance of an explicit symmetric positive product rule in H_T^*(P^n). The construction relies only on the standard localization theorem and the torus action, as described. Since the report contains no specific major comments or points of concern, we believe the explicit rule provided in the paper stands as a complete resolution of the Anderson-Fulton problem.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard tools

full rationale

The paper resolves the Anderson-Fulton problem by exhibiting an explicit symmetric positive product rule in H_T^*(P^n). The provided abstract and context indicate that the construction rests on the standard torus action on projective space together with the usual localization isomorphism to the fixed-point basis; both are external, classical results in equivariant cohomology and are not derived from the target rule itself. No equations, self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the visible material. The central claim therefore remains independent of its own inputs and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes the standard setup of torus-equivariant cohomology on P^n and the localization theorem, both standard in the field. No free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (2)
  • domain assumption Torus-equivariant cohomology ring of P^n with the standard action
    The problem statement and solution are framed inside this established geometric setting.
  • standard math Localization theorem for equivariant cohomology
    Implicitly used to reduce computations to fixed-point data.

pith-pipeline@v0.9.0 · 5287 in / 1150 out tokens · 35720 ms · 2026-05-11T01:22:50.156706+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    Anderson and W

    D. Anderson and W. Fulton.Equivariant cohomology in algebraic geometry, volume 210 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2024

    work page 2024

  2. [2]

    Anderson, E

    D. Anderson, E. Richmond, and A. Yong. Eigenvalues of Hermitian matrices and equivariant coho- mology of Grassmannians.Compos. Math., 149(9):1569–1582, 2013

    work page 2013

  3. [3]

    W. Graham. Positivity in equivariant Schubert calculus.Duke Math. J., 109(3):599–614, 2001

    work page 2001

  4. [4]

    A. J. Hoffman and J. B. Kruskal. Integral boundary points of convex polyhedra. InLinear inequalities and related systems, Ann. of Math. Stud., no. 38, pages 223–246. Princeton Univ. Press, Princeton, NJ, 1956

    work page 1956

  5. [5]

    Knutson and T

    A. Knutson and T. Tao. Puzzles and (equivariant) cohomology of Grassmannians.Duke Math. J., 119(2):221–260, 2003

    work page 2003

  6. [6]

    Monical, N

    C. Monical, N. Tokcan, and A. Yong. Newton polytopes in algebraic combinatorics.Selecta Math. (N.S.), 25(5):Paper No. 66, 37, 2019

    work page 2019

  7. [7]

    Postnikov

    A. Postnikov. Permutohedra, associahedra, and beyond.Int. Math. Res. Not. IMRN, (6):1026–1106, 2009

    work page 2009

  8. [8]

    Robichaux, H

    C. Robichaux, H. Yadav, and A. Yong. Equivariant cohomology, Schubert calculus, and edge labeled tableaux. InFacets of algebraic geometry. Vol. II, volume 473 ofLondon Math. Soc. Lecture Note Ser., pages 284–335. Cambridge Univ. Press, Cambridge, 2022

    work page 2022

  9. [9]

    Schrijver.Combinatorial optimization

    A. Schrijver.Combinatorial optimization. Polyhedra and efficiency. Vol. C, volume 24 ofAlgorithms and Combinatorics. Springer-Verlag, Berlin, 2003. Disjoint paths, hypergraphs, Chapters 70–83

    work page 2003

  10. [10]

    Thomas and A

    H. Thomas and A. Yong. Equivariant Schubert calculus and jeu de taquin.Ann. Inst. Fourier (Grenoble), 68(1):275–318, 2018. DEPT.OFMATHEMATICS, U. ILLINOIS ATURBANA-CHAMPAIGN, URBANA, IL 61801, USA Email address:duyphan2@illinois.edu

    work page 2018