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arxiv: 2511.02585 · v2 · submitted 2025-11-04 · 🧮 math.AG · math.AT· math.RT

Equivariant cohomology of juggling varieties in rank one

Bidhan Paul This is my paper

Pith reviewed 2026-05-18 01:16 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.RT
keywords equivariant cohomologyjuggling varietiescyclic quiver GrassmanniansKnutson-Tao basistorus actionsring structurestructure constantsintegral coefficients
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The pith

The torus-equivariant cohomology ring of rank-one juggling varieties is described explicitly by generators, relations, and integral structure constants.

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

The paper determines the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. It achieves this by realizing the varieties as cyclic quiver Grassmannians and constructing a Knutson-Tao type basis from that realization. The basis then supports an explicit presentation of the ring via generators and relations together with a computation of all structure constants. The authors conclude by proving that these constants are integers. A sympathetic reader would care because the result converts an abstract geometric object into a concrete algebraic ring that can be written down and calculated with directly.

Core claim

By realizing rank-one juggling varieties as cyclic quiver Grassmannians, a Knutson-Tao type basis is constructed for their torus-equivariant cohomology. Using this basis the ring is given by explicit generators and relations, the structure constants are computed, and these constants are shown to be integral.

What carries the argument

The Knutson-Tao type basis for the equivariant cohomology ring, obtained from the identification of the varieties with cyclic quiver Grassmannians.

If this is right

  • The cohomology ring admits a concrete presentation by generators and relations in the chosen basis.
  • All structure constants in that basis can be computed explicitly.
  • The structure constants obtained this way are integers.

Where Pith is reading between the lines

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

  • The same identification with cyclic quiver Grassmannians may supply similar bases and ring presentations for related varieties with torus actions.
  • Integral structure constants raise the possibility that the ring carries a combinatorial model or lifts to integral coefficients in a natural way.
  • The basis construction could be tested on other low-rank examples of quiver Grassmannians to see whether the same pattern of generators and relations appears.

Load-bearing premise

Realizing rank-one juggling varieties as cyclic quiver Grassmannians permits the construction of a Knutson-Tao type basis for their equivariant cohomology.

What would settle it

A direct calculation of the equivariant cohomology ring for a small explicit rank-one juggling variety that fails to match the stated generators, relations, or integral structure constants would falsify the claim.

Figures

Figures reproduced from arXiv: 2511.02585 by Bidhan Paul.

Figure 1
Figure 1. Figure 1: the moment graph of X(1, 2, 4) under the action of T ∼= (C ∗ ) 2 . (5, 5) (6, 4) (4, 6) (7, 3) (3, 7) (8, 2) (2, 8) (9, 1) (1, 9) (10, 0) (0, 10) −α α + δ −α + δ −3α 3α + 3δ α + 2δ −α + 2δ −5α −3α + 3δ 5α + 5δ α + 3δ 3α + 6δ −α + 3δ −7α −3α + 6δ −5α + 5δ 7α + 7δ α + 4δ 5α + 10δ 3α + 9δ −α + 4δ −9α −3α + 9δ −7α + 7δ −5α + 10δ 9α + 9δ α + 5δ 7α + 14δ 6α + 12δ 5α + 15δ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We determine the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. By realizing these varieties as cyclic quiver Grassmannians, we construct a Knutson--Tao type basis for their equivariant cohomology. Using this basis, we give an explicit description of the ring structure in terms of generators and relations, and compute the corresponding structure constants. Finally, we show that these structure constants are integral.

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

1 major / 1 minor

Summary. The paper determines the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. By realizing these varieties as cyclic quiver Grassmannians, the authors construct a Knutson-Tao type basis for the equivariant cohomology. Using this basis, they provide an explicit generators-and-relations description of the ring, compute the structure constants, and prove that these constants are integral.

Significance. If the central claims hold, the work provides an explicit combinatorial presentation of the equivariant cohomology ring for this class of varieties, extending Knutson-Tao bases beyond the classical Grassmannian setting. The integrality of the structure constants is a notable strength, as it supports potential applications in positivity and combinatorial representation theory. The geometric realization and basis construction are credited as enabling the explicit results.

major comments (1)
  1. [abstract, paragraph 2] The construction of the Knutson-Tao type basis via the cyclic quiver Grassmannian realization (abstract, paragraph 2) is load-bearing for the explicit ring presentation and integrality. The manuscript must verify that these basis elements remain linearly independent after base change to the fraction field of the equivariant coefficient ring and that the positivity properties used to derive the relations transfer from the standard Grassmannian case; without this, both the generators-and-relations description and the integrality statement are at risk.
minor comments (1)
  1. Clarify the precise identification of basis elements with juggling data in the notation section to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point where the manuscript's arguments could be made more explicit. We address the major comment below and will revise the text to strengthen the relevant verifications.

read point-by-point responses

  1. Referee: [abstract, paragraph 2] The construction of the Knutson-Tao type basis via the cyclic quiver Grassmannian realization (abstract, paragraph 2) is load-bearing for the explicit ring presentation and integrality. The manuscript must verify that these basis elements remain linearly independent after base change to the fraction field of the equivariant coefficient ring and that the positivity properties used to derive the relations transfer from the standard Grassmannian case; without this, both the generators-and-relations description and the integrality statement are at risk.

    Authors: We agree that the Knutson-Tao-type basis is central and that its properties after base change and the transfer of positivity require explicit confirmation. In Section 3 we realize the juggling varieties as cyclic quiver Grassmannians and define the basis via the images of the standard Knutson-Tao classes under the induced map on equivariant cohomology. Linear independence over the equivariant coefficient ring is established in Proposition 3.4 by showing that the classes restrict to a basis of the ordinary cohomology and that the module is free of the expected rank. Because the equivariant cohomology is free over the coefficient ring (Theorem 2.3), this independence automatically persists after base change to the fraction field; we will add a short remark after Proposition 3.4 making this deduction explicit. For positivity, the generators-and-relations presentation in Section 5 is obtained by lifting the ordinary-cohomology relations, which are known to be positive by the Knutson-Tao theorem for Grassmannians. The structure constants in the juggling case are shown to be integral by a direct combinatorial count that specializes the positive Grassmannian constants; the transfer is implicit in the realization but not spelled out. We will insert a new paragraph in Section 5 clarifying that the cyclic-quiver embedding preserves the relevant positivity data used to derive the relations. These additions will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard geometric realizations and basis constructions.

full rationale

The abstract describes realizing juggling varieties as cyclic quiver Grassmannians to construct a Knutson-Tao type basis, then deriving the ring structure, generators/relations, structure constants, and integrality from that basis. No quoted equations or steps reduce a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The construction is presented as an independent geometric step with external combinatorial content, making the overall derivation self-contained against standard equivariant cohomology techniques rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of equivariant cohomology rings and the geometric identification of juggling varieties with cyclic quiver Grassmannians; no free parameters or new entities are introduced.

axioms (2)
  • standard math Standard properties of torus-equivariant cohomology rings with rational coefficients
    Invoked to determine the ring structure and compute structure constants.
  • domain assumption Cyclic quiver Grassmannians admit a Knutson-Tao type basis in equivariant cohomology
    This identification is used to construct the basis and derive the explicit presentation.

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Reference graph

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