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arxiv: 2507.15183 · v2 · pith:JTBRITYNnew · submitted 2025-07-21 · 🧮 math.AG · math.CO

A Nakayama result for the quantum K theory of homogeneous spaces

Wei Gu , Leonardo C. Mihalcea , Eric Sharpe , Weihong Xu , Hao Zhang , Hao Zou This is my paper

Pith reviewed 2026-05-19 04:45 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords quantum K theoryhomogeneous spacesideal of relationsquantum deformationpartial flag manifoldsequivariant K ringWhitney relations
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The pith

The ideal of relations in the quantum K ring of a homogeneous space is generated by quantizations of the classical generators.

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

This paper establishes that the defining relations for the quantum K-ring of homogeneous spaces arise by taking quantizations of the known generators of the classical K-ring ideal. A sympathetic reader would care because this supplies an explicit way to produce the full set of quantum relations from classical data without needing independent quantum corrections at the relation level. The claim covers both ordinary and equivariant versions of the rings. It is illustrated for partial flag manifolds by invoking a set of quantum K Whitney relations that generate the ideal in that case.

Core claim

The ideal of relations in the (equivariant) quantum K ring of a homogeneous space is generated by quantizations of each of the generators of the ideal in the classical (equivariant) K ring.

What carries the argument

The natural map from the classical K-ring to its quantum deformation that sends each classical generator of the relation ideal to its quantization, with those images generating the full quantum relation ideal.

If this is right

  • The quantum K-ring for any homogeneous space can be presented by quantizing the classical generators rather than deriving new independent relations.
  • The same generation property holds in the equivariant setting for both the quantum and classical rings.
  • For partial flag manifolds the quantum K Whitney relations suffice to generate the entire ideal once quantized.
  • Any computation that starts from a classical presentation of the K-ring ideal immediately yields a presentation of the quantum version.

Where Pith is reading between the lines

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

  • The result would let researchers obtain explicit bases or structure constants for quantum K-theory on homogeneous spaces by lifting classical computations.
  • Analogous generation statements might hold when the base variety is replaced by other spaces whose classical K-rings are known by generators and relations.
  • Direct verification on low-dimensional examples such as projective spaces or Grassmannians would give concrete checks that the quantized generators indeed span all quantum relations.

Load-bearing premise

The quantum K-ring is constructed so that its ideal of relations is generated precisely by the images of the quantized classical generators under the natural map from the classical ring.

What would settle it

Exhibiting one homogeneous space together with an explicit quantum relation that cannot be written as a polynomial combination of the quantized classical generators would falsify the claim.

read the original abstract

We prove that the ideal of relations in the (equivariant) quantum K ring of a homogeneous space is generated by quantizations of each of the generators of the ideal in the classical (equivariant) K ring. This extends to quantum K theory a result of Siebert and Tian in quantum cohomology. We illustrate this technique in the case of the quantum K ring of partial flag manifolds, using a set of quantum K Whitney relations conjectured by the authors, and recently proved by Huq-Kuruvilla.

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 proves that the ideal of relations in the (equivariant) quantum K-ring of a homogeneous space is generated by quantizations of the generators of the corresponding ideal in the classical (equivariant) K-ring. This extends the Siebert-Tian result from quantum cohomology to the K-theoretic setting and illustrates the method for partial flag manifolds by invoking a set of quantum K-Whitney relations that were conjectured by the authors and proved by Huq-Kuruvilla.

Significance. If the central claim holds, the result supplies a structural tool that reduces the problem of presenting quantum K-rings of homogeneous spaces to the deformation of known classical relations. It thereby connects directly to existing computations in quantum cohomology and K-theory and leverages an external verification of the Whitney relations for the flag-manifold case.

major comments (1)
  1. [Proof of the main theorem] The proof of the main statement proceeds by reducing modulo the augmentation ideal (q=0) to recover the classical generators via Siebert-Tian and then invoking a Nakayama-type lifting argument. In the equivariant setting the ring is defined over a completion such as K_T[[q]] or a Novikov ring; the manuscript does not explicitly verify that the module of relations remains flat (or torsion-free) over this base. Without such a verification, the lifting step does not automatically guarantee that no additional generators appear at positive order in q.
minor comments (1)
  1. [Abstract and §1] The abstract and introduction should clarify whether the result applies to the completed or the localized quantum K-ring, as the choice affects the flatness hypothesis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for your detailed report and for recognizing the significance of extending the Siebert-Tian theorem to the quantum K-theoretic setting. We appreciate the suggestion to strengthen the proof in the equivariant case. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [Proof of the main theorem] The proof of the main statement proceeds by reducing modulo the augmentation ideal (q=0) to recover the classical generators via Siebert-Tian and then invoking a Nakayama-type lifting argument. In the equivariant setting the ring is defined over a completion such as K_T[[q]] or a Novikov ring; the manuscript does not explicitly verify that the module of relations remains flat (or torsion-free) over this base. Without such a verification, the lifting step does not automatically guarantee that no additional generators appear at positive order in q.

    Authors: We thank the referee for highlighting this subtlety in the equivariant setting. The proof indeed relies on a Nakayama-type argument after reducing modulo the ideal (q). While the manuscript applies this in the completed ring K_T[[q]], we acknowledge that an explicit check of flatness of the relation module over this base was not included. In fact, the equivariant K-ring of a homogeneous space is free as a module over the coefficient ring K_T, and the quantum deformation preserves this freeness because the relations are deformed in a way that they remain a regular sequence in the formal power series ring. Consequently, the module of relations is free (hence flat and torsion-free) over K_T[[q]]. To make this rigorous, we will add a short paragraph in Section 2 or the proof of the main theorem explaining this flatness, perhaps by citing the known freeness in classical equivariant K-theory and noting that the quantization does not introduce torsion as the leading terms generate the classical ideal. This will ensure that the lifting produces all relations without additional generators at positive q-orders. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim proved by lifting from external classical result via Nakayama argument

full rationale

The paper's main theorem extends the Siebert-Tian result on quantum cohomology to quantum K-theory by applying a Nakayama-type lifting argument to the ideal of relations. The proof reduces modulo q=0 to recover the classical case (which is external) and then lifts generators; this does not reduce to a self-definition or fitted input inside the paper. The illustration for partial flags invokes the Huq-Kuruvilla proof of the authors' conjectured Whitney relations, which is an independent external verification. No load-bearing step equates the quantum relation ideal to its classical image by construction, and the ring is treated as a deformation over a completed base without internal fitting of the generators themselves. The derivation is therefore self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of the quantum K-ring and on the recent external proof of the quantum K Whitney relations; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The quantum K-ring of a homogeneous space is defined via the standard quantum product that deforms the classical K-ring.
    This is the conventional setup in quantum K-theory as referenced in the abstract.

pith-pipeline@v0.9.0 · 5617 in / 1103 out tokens · 61855 ms · 2026-05-19T04:45:33.333169+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

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  1. Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials

    hep-th 2026-01 unverdicted novelty 7.0

    Schubert line defects in 3d GLSMs for partial flag manifolds reproduce parabolic Whitney polynomials for Schubert classes in quantum K-theory and yield new parabolic quantum Grothendieck polynomials.

  2. Schubert line defects in 3d GLSMs, part I: Complete flag manifolds and quantum Grothendieck polynomials

    hep-th 2025-12 unverdicted novelty 7.0

    Schubert line defects in 3d GLSMs for complete flag manifolds are realized as SQM quivers whose indices give quantum Grothendieck polynomials and restrict the target space to Schubert varieties.

  3. Total instanton restriction via multiverse interference: Noncompact gauge theories and (-1)-form symmetries

    hep-th 2025-07 unverdicted novelty 6.0

    Continuous-universe decomposition plus (-1)-form gauging eliminates every instanton in local QFTs, realized explicitly by switching 2D U(1) gauge theories to noncompact R gauge groups.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 3 Pith papers

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    [Eis95] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol

    MR1932326 [CG09] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry , Springer Science & Business Media, 2009. [Eis95] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR1322960 [FL94] William Fulton and Alain Lascoux, A Pieri form...

    work page 2009

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    Quantum K Whitney relations for partial flag varieties,

    Dedicated to William Fulton on the occasion of his 60th birthday. MR1786492 (2001m:14078) [GK17] Vassily Gorbounov and Christian Korff, Quantum integrability and generalised quantum Schubert cal- culus, Adv. Math. 313 (2017), 282–356. MR3649227 [GK24] Wei Gu and Elana Kalashnikov, A rim-hook rule for quiver flag varieties , Selecta Math. (N.S.) 30 (2024),...

    work page arXiv 2017

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    MR4719974 14 WEI GU, LEONARDO C. MIHALCEA, ERIC SHARPE, WEIHONG XU, HAO ZHANG, AND HAO ZOU Zhejiang Institute of Modern Physics, School of Physics, Zhejiang University, Hangzhou, Zhejiang 310058, China Email address: guwei2875@zju.edu.cn Department of Mathematics, 225 Stanger Street, McBryde Hall, Virginia Tech University, Blacks- burg, V A 24061 USA Emai...

    work page