Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Daewoong Cheong; Manwook Han

arxiv: 1906.11646 · v1 · pith:WD6CSMKBnew · submitted 2019-06-26 · 🧮 math.AG

Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Daewoong Cheong , Manwook Han This is my paper

Pith reviewed 2026-05-25 15:24 UTC · model grok-4.3

classification 🧮 math.AG

keywords Galkin conjecturequantum cohomologyLagrangian Grassmannianorthogonal GrassmannianFano manifoldeigenvalue boundfirst Chern class

0 comments

The pith

Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians except in some equality cases.

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

This paper proves that for Lagrangian and orthogonal Grassmannians the maximal real eigenvalue δ0 of quantum multiplication by the first Chern class satisfies δ0 at least equal to the manifold dimension plus one. The result follows from the known existence of this maximal eigenvalue for these spaces together with direct verification of the inequality. Equality holds precisely when the manifold is projective space, except for the noted exceptions listed in the paper. A sympathetic reader cares because the statement gives a uniform lower bound on a quantum-cohomology invariant across an infinite family of Fano varieties. The argument extends earlier verifications to these two classical homogeneous spaces.

Core claim

The paper shows that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

What carries the argument

The quantum multiplication operator [c1(M)] by the first Chern class together with its maximal real eigenvalue δ0.

If this is right

The inequality δ0 ≥ dim M + 1 holds for every Lagrangian Grassmannian and every orthogonal Grassmannian.
Equality δ0 = dim M + 1 occurs only when the variety is projective space, subject to the listed exceptions.
The conjecture is therefore verified for these two infinite families of Fano manifolds.
The same maximal-eigenvalue machinery already known for these spaces suffices to establish the bound.

Where Pith is reading between the lines

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

The same verification technique might apply to other homogeneous spaces once the existence of δ0 is known.
Classifying the exceptional equality cases more precisely could reveal whether they share a common geometric feature.
The bound supplies a numerical constraint that mirror-symmetric computations on these Grassmannians must satisfy.

Load-bearing premise

The operator [c1(M)] possesses a real-valued maximal eigenvalue δ0, a property already established for these Grassmannians in prior works.

What would settle it

An explicit computation of the spectrum of [c1(M)] on any Lagrangian or orthogonal Grassmannian yielding a maximal real eigenvalue strictly less than dim M + 1.

read the original abstract

Let $M$ be a Fano manifold, and $H^\star(M;\mathbb{C})$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in H^*(M;\mathbb{C})$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^*(M;\mathbb{C})$. It was conjectured several years ago \cite{GGI, GI} and has been proved for many Fano manifols \cite{CL1, CH2, LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real valued eigenvalue $\delta_0$ which is maximal among eigenvaules of $[c_1(M)]$. Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$ $\delta_0\geq \mathrm{dim} \ M +1,$ and the equlity holds if and only if $M$ is the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

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.

Desk Editor's Note private letter to a colleague

This note verifies Galkin's lower bound for Lagrangian and orthogonal Grassmannians by citing prior eigenvalue results.

read the letter

The main thing here is a verification that Galkin's lower bound conjecture holds for Lagrangian Grassmannians and orthogonal Grassmannians. The authors use the established existence of a maximal real eigenvalue δ0 for the quantum multiplication operator by the first Chern class, then show the inequality δ0 ≥ dim M + 1 with equality only in the projective space case, except for some listed exceptions. This adds two families to the known cases. The work is mostly in applying the general result to these specific varieties, which are homogeneous Fano manifolds. The paper does a clean job of stating the conjecture and the prior results it relies on. The citations are listed explicitly, and the claim is presented without overstatement. The soft spot is the dependence on those citations for the eigenvalue property. The note says the property has been proved including for these cases, but does not include an independent check or computation of the characteristic polynomial or the quantum ring structure. For a reader who wants to be sure the references apply exactly to LG(n,2n) and OG(n,2n+1), that step is missing. This is not a fatal issue if the citations are accurate, but it makes the paper more of an application than a self-contained argument. No new general technique appears. The argument follows the standard path once δ0 is granted. This paper is for people already working on Galkin's conjecture or quantum cohomology of homogeneous spaces. A reader compiling verified examples or looking for the status on Grassmannians would get value from it. It is solid enough on its own terms to deserve a serious referee, mainly to double-check the coverage of the cited works and the handling of the exceptions. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that Galkin's lower bound conjecture holds for Lagrangian Grassmannians LG(n,2n) and orthogonal Grassmannians OG(n,2n+1). It invokes the established fact (proved in the cited works GGI, GI, CL1, CH2, LiMiSh, Ke, including these cases) that the quantum multiplication operator [c1(M)] admits a real maximal eigenvalue δ0, then asserts that δ0 ≥ dim M + 1 with equality if and only if M is projective space, modulo some exceptions for the equality case.

Significance. If the central claim is correct, the note supplies further verification of Galkin's conjecture on two families of homogeneous Fano manifolds whose quantum cohomology rings are explicitly known. The argument does not introduce new computations of the quantum product or the characteristic polynomial of [c1(M)], but instead applies the spectral property already established in the literature.

major comments (1)

[Abstract] Abstract: the assertion that the maximal-eigenvalue property 'has been proved ... including our cases' for LG(n,2n) and OG(n,2n+1) supplies no explicit theorem numbers or propositions from the cited references (GGI, GI, CL1, CH2, LiMiSh, Ke) that cover these varieties. Because the subsequent comparison δ0 ≥ dim M + 1 rests directly on the existence and reality of this eigenvalue, the coverage must be documented to support the central claim.

minor comments (2)

[Title] Title: 'Conjecure' is a typographical error and should read 'Conjecture'.
[Abstract] Abstract: 'manifols' should read 'manifolds'; 'equlity' should read 'equality'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the suggestion to strengthen the abstract. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the maximal-eigenvalue property 'has been proved ... including our cases' for LG(n,2n) and OG(n,2n+1) supplies no explicit theorem numbers or propositions from the cited references (GGI, GI, CL1, CH2, LiMiSh, Ke) that cover these varieties. Because the subsequent comparison δ0 ≥ dim M + 1 rests directly on the existence and reality of this eigenvalue, the coverage must be documented to support the central claim.

Authors: We agree that the abstract would benefit from explicit theorem or proposition numbers from the cited works. In the revised version we will insert the precise references (e.g., the relevant statements in GGI, GI, CL1, CH2, LiMiSh and Ke) that establish the existence and reality of the maximal eigenvalue δ0 for LG(n,2n) and OG(n,2n+1). This documentation will make the logical foundation of the subsequent inequality fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external citations as independent input

full rationale

The paper states that the existence of the real maximal eigenvalue δ0 for the operator [c1(M)] 'has been proved for many Fano manifolds including our cases' via citations, then uses this to verify Galkin's inequality δ0 ≥ dim M + 1 for the specified Grassmannians. No equation or claim in the provided text reduces the target result to a fitted parameter, self-definition, or prior result by the same authors in a closed loop. The eigenvalue property is treated as an external premise rather than derived from the conjecture itself, and the application to Lagrangian/orthogonal Grassmannians supplies separate content. This matches the default case of a non-circular paper whose central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the prior establishment of the maximal real eigenvalue δ0 for the manifolds under study and on standard properties of quantum cohomology rings of Fano manifolds.

axioms (1)

domain assumption The quantum cohomology ring of a Fano manifold admits a well-defined quantum product and the operator [c1(M)] possesses a real maximal eigenvalue δ0.
Invoked in the abstract via citations to GGI, GI and subsequent proofs.

pith-pipeline@v0.9.0 · 5740 in / 1272 out tokens · 28199 ms · 2026-05-25T15:24:40.849453+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

It was conjectured several years ago [GGI, GI] ... that the operator [c1(M)] has a real valued eigenvalue δ0 which is maximal among eigenvalues of [c1(M)]. ... δ0(LG(n)) = 2^{-1/(n+1)}(n+1)(sin π/(2(n+1)))^{-1}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Galkin’s lower bound conjecture states that for a Fano manifold M, δ0 ≥ dim M + 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 3 internal anchors

[1]

Cheong, Quantum multiplication operators for Lagrangian and ortho gonal Grassmannians , J

D. Cheong, Quantum multiplication operators for Lagrangian and ortho gonal Grassmannians , J. Al- gbraic Combin. 45 (4), pp. 1153-1171, 2017

work page 2017
[2]

Cheong and C

D. Cheong and C. Li, On the Conjecture O of Galkin, Golyshev and Iritani for G/P , Advances in Math., 306 (2017) pp. 704-721

work page 2017
[3]

Evans, L

L. Evans, L. Schneider, R. Shiﬂer, L. Short and S. Warman, Galkin ’s lower conjecture holds for the Grassmannian, preprint

work page
[4]

Fulton, Young Tableaux, London Mathematical Society, Cambridge Univ

W. Fulton, Young Tableaux, London Mathematical Society, Cambridge Univ. Press, Camb ridge, 1997

work page 1997
[5]

Fulton and C

W. Fulton and C. Woodward, On the quantum product of Schubert classes , J. Algebraic Geom. 13 (2004), no. 4, 641-661

work page 2004
[6]

Galkin and V

S. Galkin and V. Golyshev, Quantum cohomology of Grassmannians and cyclotomic ﬁelds , Russian Mathematical Surveys (2006), 61(1):171

work page 2006
[7]

Galkin, V

S. Galkin, V. Golyshev and H. Iritani, Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures, Duke Math. J. 165 (2016), no. 11, 2005-2077

work page 2016
[8]

The conifold point

S. Galkin, The conifold point , arXiv:1404.7388

work page internal anchor Pith review Pith/arXiv arXiv
[9]

Galkin and H

S. Galkin and H. Iritani, Gamma conjectures and mirror symmetry , arXiv:1508.00719v2

work page arXiv
[10]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-Verlag New York Inc., 1972

work page 1972
[11]

On Conjecture $\mathcal O$ for projective complete intersections

H. Ke, On Conjecture O for projective complete intersections , arXiv: 1809.10869

work page internal anchor Pith review Pith/arXiv arXiv
[12]

Kresch and H

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian , J. Algebraic Ge- ometry 12 (2003), 777-810. 10 DAEWOONG CHEON AND MANWOOK HAN

work page 2003
[13]

Kresch and H

A. Kresch and H. Tamvakis, Quantum cohomology of orthogonal Grassmannians , Compositio Math. 140 (2004), 482-500

work page 2004
[14]

C. Li, L. Mihalcea and R. Shiﬂer, Conjecture O holds for the odd symplectic Grassmannian , arXiv:1706.00744

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Lascoux and P

A. Lascoux and P. Pragacz, Operator calculus for ˜Q-Polynomials and Schubert polynomials , Adv. in Math 140 (1998), 1-43

work page 1998
[16]

Macdonald, Symmetric Functions and Hall Polynomials , Second edition, Oxford Univ

I.G. Macdonald, Symmetric Functions and Hall Polynomials , Second edition, Oxford Univ. Press, 1995

work page 1995
[17]

Pragacz and J

P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; ˜Q-polynomial approach, Compositio Mathematica 107; 11-87, 1997

work page 1997
[18]

Rietsch, Quantum cohomology rings of Grassmannians and total positi vity, Duke Mathematical Journal, Vol 110, no

K. Rietsch, Quantum cohomology rings of Grassmannians and total positi vity, Duke Mathematical Journal, Vol 110, no. 3 (2001), 523-553. Chungbuk National University, Department of Mathematics, Chungdae-ro 1, Seowon-Gu, Cheongju City, Chungbuk 28644, Korea E-mail address : daewoongc@chungbuk.ac.kr Chungbuk National University, Department of Mathematics, Ch...

work page 2001

[1] [1]

Cheong, Quantum multiplication operators for Lagrangian and ortho gonal Grassmannians , J

D. Cheong, Quantum multiplication operators for Lagrangian and ortho gonal Grassmannians , J. Al- gbraic Combin. 45 (4), pp. 1153-1171, 2017

work page 2017

[2] [2]

Cheong and C

D. Cheong and C. Li, On the Conjecture O of Galkin, Golyshev and Iritani for G/P , Advances in Math., 306 (2017) pp. 704-721

work page 2017

[3] [3]

Evans, L

L. Evans, L. Schneider, R. Shiﬂer, L. Short and S. Warman, Galkin ’s lower conjecture holds for the Grassmannian, preprint

work page

[4] [4]

Fulton, Young Tableaux, London Mathematical Society, Cambridge Univ

W. Fulton, Young Tableaux, London Mathematical Society, Cambridge Univ. Press, Camb ridge, 1997

work page 1997

[5] [5]

Fulton and C

W. Fulton and C. Woodward, On the quantum product of Schubert classes , J. Algebraic Geom. 13 (2004), no. 4, 641-661

work page 2004

[6] [6]

Galkin and V

S. Galkin and V. Golyshev, Quantum cohomology of Grassmannians and cyclotomic ﬁelds , Russian Mathematical Surveys (2006), 61(1):171

work page 2006

[7] [7]

Galkin, V

S. Galkin, V. Golyshev and H. Iritani, Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures, Duke Math. J. 165 (2016), no. 11, 2005-2077

work page 2016

[8] [8]

The conifold point

S. Galkin, The conifold point , arXiv:1404.7388

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Galkin and H

S. Galkin and H. Iritani, Gamma conjectures and mirror symmetry , arXiv:1508.00719v2

work page arXiv

[10] [10]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-Verlag New York Inc., 1972

work page 1972

[11] [11]

On Conjecture $\mathcal O$ for projective complete intersections

H. Ke, On Conjecture O for projective complete intersections , arXiv: 1809.10869

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

Kresch and H

A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian , J. Algebraic Ge- ometry 12 (2003), 777-810. 10 DAEWOONG CHEON AND MANWOOK HAN

work page 2003

[13] [13]

Kresch and H

A. Kresch and H. Tamvakis, Quantum cohomology of orthogonal Grassmannians , Compositio Math. 140 (2004), 482-500

work page 2004

[14] [14]

C. Li, L. Mihalcea and R. Shiﬂer, Conjecture O holds for the odd symplectic Grassmannian , arXiv:1706.00744

work page internal anchor Pith review Pith/arXiv arXiv

[15] [15]

Lascoux and P

A. Lascoux and P. Pragacz, Operator calculus for ˜Q-Polynomials and Schubert polynomials , Adv. in Math 140 (1998), 1-43

work page 1998

[16] [16]

Macdonald, Symmetric Functions and Hall Polynomials , Second edition, Oxford Univ

I.G. Macdonald, Symmetric Functions and Hall Polynomials , Second edition, Oxford Univ. Press, 1995

work page 1995

[17] [17]

Pragacz and J

P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; ˜Q-polynomial approach, Compositio Mathematica 107; 11-87, 1997

work page 1997

[18] [18]

Rietsch, Quantum cohomology rings of Grassmannians and total positi vity, Duke Mathematical Journal, Vol 110, no

K. Rietsch, Quantum cohomology rings of Grassmannians and total positi vity, Duke Mathematical Journal, Vol 110, no. 3 (2001), 523-553. Chungbuk National University, Department of Mathematics, Chungdae-ro 1, Seowon-Gu, Cheongju City, Chungbuk 28644, Korea E-mail address : daewoongc@chungbuk.ac.kr Chungbuk National University, Department of Mathematics, Ch...

work page 2001