Gamma conjecture I for flag varieties

Chi Hong Chow

arxiv: 2501.13221 · v3 · submitted 2025-01-22 · 🧮 math.AG · math-ph· math.MP· math.RT

Gamma conjecture I for flag varieties

Chi Hong Chow This is my paper

Pith reviewed 2026-05-23 05:17 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MPmath.RT

keywords gamma conjectureflag varietiesmirror symmetryrietsch mirrorhat-gamma classperron-frobenius eigenvaluequantum cohomology

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The pith

Gamma conjecture I holds for all flag varieties.

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

The paper proves that Gamma conjecture I is true for every flag variety. It does so by adapting the strategy of Galkin and Iritani, where the main addition is verifying that the totally positive part of the Rietsch mirror corresponds to the hat-Gamma class and includes the critical point of the superpotential for the Perron-Frobenius eigenvalue. A reader might care because this links the A-model and B-model sides of mirror symmetry for these varieties. The result confirms a key prediction in the field for this large family of spaces.

Core claim

We prove Gamma conjecture I for all flag varieties by following a strategy proposed by Galkin and Iritani. The main new ingredient is showing that the totally positive part of the Rietsch mirror is mirror to the hat-Gamma-class and contains the critical point of the superpotential that corresponds to the Perron-Frobenius eigenvalue on the A-side.

What carries the argument

The totally positive part of the Rietsch mirror, shown to be mirror to the hat-Gamma-class while containing the superpotential critical point for the Perron-Frobenius eigenvalue.

If this is right

Gamma conjecture I holds for all flag varieties.
The totally positive part of the Rietsch mirror is mirror to the hat-Gamma-class.
This part of the mirror contains the critical point corresponding to the Perron-Frobenius eigenvalue on the A-side.
The Galkin-Iritani strategy extends to flag varieties.

Where Pith is reading between the lines

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

Similar mirror verifications could apply to other classes of varieties with known mirrors.
The result may aid in computing the Gamma class in quantum cohomology settings for flag varieties.
Explicit checks on low-rank flag varieties could test the critical point location numerically.

Load-bearing premise

The Galkin and Iritani strategy applies directly to flag varieties and the verification that the totally positive Rietsch mirror contains the relevant critical point holds without extra restrictions.

What would settle it

Observation of a flag variety where the totally positive part of the Rietsch mirror does not contain the critical point of the superpotential for the Perron-Frobenius eigenvalue.

read the original abstract

We prove Gamma conjecture I for all flag varieties by following a strategy proposed by Galkin and Iritani. The main new ingredient is showing that the totally positive part of the Rietsch mirror is mirror to the $\widehat{\Gamma}$-class and contains the critical point of the superpotential that corresponds to the Perron-Frobenius eigenvalue on the A-side.

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

Chow proves Gamma conjecture I for all flag varieties by verifying the Rietsch mirror condition under the Galkin-Iritani strategy.

read the letter

The main thing to know is that this paper claims a complete proof of Gamma conjecture I for every flag variety. It follows the strategy laid out by Galkin and Iritani and adds one new geometric check: that the totally positive part of the Rietsch mirror is mirror to the hat-Gamma class and contains the critical point of the superpotential matching the Perron-Frobenius eigenvalue on the A-side. That verification step is presented as the central addition, extending earlier partial results to the full class without extra restrictions on the varieties. The argument is organized cleanly around this single new ingredient, and the abstract states the claim directly. The paper does a solid job of tying the mirror symmetry statements together for this family of homogeneous spaces, which should help with related computations in quantum cohomology and K-theory. The soft spots are limited. The result stands or falls on whether the Rietsch mirror verification holds uniformly, and while the description gives no sign of hidden assumptions or inconsistencies, that step would need close reading in the full text to confirm there are no gaps. There is no circularity or reliance on fitted quantities visible here. This is for readers already working on mirror symmetry, quantum cohomology, or K-theory of flag varieties and other homogeneous spaces. A serious referee should see it because the claim is a full proof for an important class and the method is grounded in prior work with one focused extension. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper proves Gamma conjecture I for all flag varieties by following the Galkin-Iritani strategy. The central new ingredient is the verification that the totally positive part of the Rietsch mirror is mirror to the hat-Gamma class and contains the critical point of the superpotential corresponding to the Perron-Frobenius eigenvalue on the A-side.

Significance. If the result holds, this completes a uniform proof of Gamma conjecture I for the entire class of flag varieties, extending the Galkin-Iritani framework with a geometric check on the Rietsch mirror that directly links the hat-Gamma class to the totally positive locus. The argument supplies a falsifiable mirror correspondence without introducing free parameters or ad-hoc reductions.

minor comments (2)

§1, paragraph 3: the statement that the strategy 'applies directly' would benefit from a one-sentence recap of the precise hypotheses from Galkin-Iritani that are being invoked for flag varieties.
Notation: the hat on Gamma is introduced in the abstract but first defined only in §3; a forward reference in the introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report accurately summarizes the main contribution.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states it proves Gamma conjecture I for flag varieties by following the external Galkin-Iritani strategy, with the sole new ingredient being an independent geometric verification that the totally positive part of the Rietsch mirror is mirror to the hat-Gamma class and contains the relevant critical point. No derivation step is shown to reduce by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim rests on an external method plus a fresh check that does not presuppose the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard results in algebraic geometry and mirror symmetry; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard axioms of algebraic geometry and properties of quantum cohomology rings for flag varieties
Invoked implicitly as background for the conjecture and mirror constructions.

pith-pipeline@v0.9.0 · 5571 in / 1129 out tokens · 37209 ms · 2026-05-23T05:17:48.177103+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.

We prove Gamma conjecture I for any flag varieties by following a strategy proposed by Galkin and Iritani. The main ingredient is to prove that the ˆΓ-class of a flag variety is mirror to the totally positive part of the corresponding Rietsch mirror.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.6 ... S(ℏ,h,τ(t))(ℏ−μℏc1 ˆΓG∨/P∨) = ℏ−ℓ/2 ∑v∈WP (∫(XP,t)>0 e−fP,t/ℏ γh/ℏP,t Φ−1mir(σv)(−ℏ,h,t)) σv

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Exponential concentration for quantum periods via mirror symmetry
math.AG 2026-05 unverdicted novelty 4.0

Modified hypergeometric series respect the exponential concentration property, implying the same for quantum periods of Fano manifolds admitting convenient weak Landau-Ginzburg models with non-negative coefficients.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Baumann, J

P . Baumann, J. Kamnitzer and A. Knuston, The Mirkovi´ c-Vilonen basis and Duistermaat- Heckman measures. With an appendix by A. Dranowski, J. Kamni tzer and C. Morton- Ferguson, Acta Math. 227 (2021), no. 1, 1-101

work page 2021
[2]

Berenstein and D

A. Berenstein and D. Kazhdan, Geometric and unipotent crystals , Geom. Funct. Anal., Spe- cial V olume, Part I (2000), 188-236

work page 2000
[3]

Berenstein and D

A. Berenstein and D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicry s- tals to crystal bases , in: Quantum groups, Contemp. Math. 433, American Mathematical Society, Providence, RI, 2007, 13-88

work page 2007
[4]

Berenstein and A

A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totall y positive varieties, Invent. Math. 143 (2001), no. 1, 77-128

work page 2001
[5]

Berenstein and A

A. Berenstein and A. Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128-166

work page 1997
[6]

Cheong and C

D. Cheong and C. Li, On the Conjecture O of GGI forG/P , Adv. Math. 306 (2017), 704-721

work page 2017
[7]

Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion

R. Chhaibi, Littelmann path model for geometric crystals, Whittaker fu nctions on Lie groups and Brownian motion, Ph.D. Thesis, available at https://arxiv.org/abs/1302.0902. 46 CHI HONG CHOW

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Chow, TheDℏ-module mirror conjecture for ﬂag varieties , Preprint (2023), available at https://arxiv.org/abs/2311.15523

C.H. Chow, TheDℏ-module mirror conjecture for ﬂag varieties , Preprint (2023), available at https://arxiv.org/abs/2311.15523

work page arXiv 2023
[9]

Cox and S

D. Cox and S. Katz, Mirror symmetry and algebraic geometr y, in: Mathematical surveys and monographs, 68, American Mathematical Society, Providence, RI, 1999

work page 1999
[10]

Galkin, J

S. Galkin, J. Hu, H. Iritani, H. Ke, C. Li and Z. Su, Revisiting Gamma conjecture I: coun- terexamples and modiﬁcations, Preprint (2024), available at https://arxiv.org/abs/2405.16979

work page arXiv 2024
[11]

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
[12]

Galkin and H

S. Galkin and H. Iritani, Gamma conjecture via mirror symmetry , in: Primitive forms and related subjects- Kavli IPMU 2014, Adv. Stud. Pure Math., 83, Math. Soc. Japan, 2019, 55- 115

work page 2014
[13]

Gerasimov, S

A. Gerasimov, S. Kharchev, D. Lebedev and S. Oblezin, On a Gauss-Givental representation of quantum Toda chain wave function , Int. Math. Res. Not. IMRN, Art. ID 96489, 1-23

work page
[14]

Gerasimov, D

A. Gerasimov, D. Lebedev and S. Oblezin, Baxter operator and Archimedean Hecke algebra , Comm. Math. Phys. 284 (2008), no. 3, 867-896

work page 2008
[15]

Gerasimov, D

A. Gerasimov, D. Lebedev and S. Oblezin, New integral representations of Whittaker func- tions for classical Lie groups , Russ. Math. Surv. 67 (2012), no. 1, 1-92

work page 2012
[16]

Givental, Stationary phase integrals, quantum Toda lattices, ﬂag man ifolds and the mir- ror conjecture, in: Topics in singularity theory: V

A. Givental, Stationary phase integrals, quantum Toda lattices, ﬂag man ifolds and the mir- ror conjecture, in: Topics in singularity theory: V . I. Arnold’s 60th anniv ersary collection, American Mathematical Society Translations: Series 2, 180, American Mathematical Soci- ety, Providence, RI, 1997, 103-115

work page 1997
[17]

Golyshev and D

V . Golyshev and D. Zagier, Proof of the gamma conjecture for Fano 3-folds of Picard rank 1, Izv. Ross. Akad. Nauk Ser. Mat., 80 (2016), no. 1, 27-54

work page 2016
[18]

Gross, Tropical geometry and mirror symmetry, in: CB MS regional conference series in Mathematics, 114, American Mathematical Society, Providence, RI, 2011

M. Gross, Tropical geometry and mirror symmetry, in: CB MS regional conference series in Mathematics, 114, American Mathematical Society, Providence, RI, 2011

work page 2011
[19]

J. Hu, H. Ke, C. Li and T. Y ang, Gamma conjecture I for del Pezzo surfaces , Adv. Math. 386 (2021)

work page 2021
[20]

Hugtenburg, On the quantum differential equations for a family of non-Kä hler monotone symplectic manifolds, Preprint (2024), available at https://arxiv.org/abs/2402.10867

K. Hugtenburg, On the quantum differential equations for a family of non-Kä hler monotone symplectic manifolds, Preprint (2024), available at https://arxiv.org/abs/2402.10867

work page arXiv 2024
[21]

Iritani, An integral structure in quantum cohomology and mirror symm etry for toric orb- ifolds, Adv

H. Iritani, An integral structure in quantum cohomology and mirror symm etry for toric orb- ifolds, Adv. Math. 222 (2009), no. 3, 1016-1079

work page 2009
[22]

Kashiwara, The crystal base and Littelmann’s reﬁned Demazure characte r formula, Duke Math

M. Kashiwara, The crystal base and Littelmann’s reﬁned Demazure characte r formula, Duke Math. J. 71 (1993), no. 3, 839-858. GAMMA CONJECTURE I FOR FLAG V ARIETIES 47

work page 1993
[23]

Kim, Quantum cohomology of ﬂag manifolds G/B and quantum Toda lattices , Ann

B. Kim, Quantum cohomology of ﬂag manifolds G/B and quantum Toda lattices , Ann. of Math. (2) 149 (1999), no. 1, 129-148

work page 1999
[24]

Knutson, T

A. Knutson, T. Lam and D.E. Speyer, Projections of Richardson varieties , J. Reine Angew. Math. 687 (2014), 133-157

work page 2014
[25]

Kumar and M

S. Kumar and M. Nori, Positivity of the cup product in cohomology of ﬂag varieties associated to Kac-Moody groups, Int. Math. Res. Not. IMRN 1998 (1998), no. 14, 757-763

work page 1998
[26]

Lam, Whittaker functions, geometric crystals, and quantum Schubert calculus, in: Schubert calculus - Osaka 2012, Adv

T. Lam, Whittaker functions, geometric crystals, and quantum Schubert calculus, in: Schubert calculus - Osaka 2012, Adv. Stud. Pure Math., 71, Math. Soc. Japan, 2016, 211-250

work page 2012
[27]

Lam and K

T. Lam and K. Rietsch, Total positivity, Schubert positivity, and geometric Sata ke, J. Algebra 460 (2016), 284-319

work page 2016
[28]

Lam and M

T. Lam and M. Shimozono, Quantum cohomology ofG/P and homology of afﬁne Grassman- nian, Acta Math. 204 (2010), no. 1, 49-90

work page 2010
[29]

Lam and N

T. Lam and N. Templier, The mirror conjecture for minuscule ﬂag varieties , Duke Math. J. 173 (2024), no. 1, 75-175

work page 2024
[30]

C. Li, K. Rietsch, M. Y ang and C. Zhang, A Plücker coordinate mirror for partial ﬂag varieties and quantum Schubert calculus , Preprint (2024), available at https://arxiv.org/abs/2401.15640

work page arXiv 2024
[31]

Lusztig, Total positivity in reductive groups , in: Lie Theory and Geometry: In Honor of Bertram Kostant, Progr

G. Lusztig, Total positivity in reductive groups , in: Lie Theory and Geometry: In Honor of Bertram Kostant, Progr. in Math. 123, Birkhäuser, 1994, 531-568

work page 1994
[32]

Manin, Frobenius manifolds, quantum cohomology and moduli spaces, in: Amer

Y . Manin, Frobenius manifolds, quantum cohomology and moduli spaces, in: Amer. Math. Soc. Colloq. Publ. 47, American Mathematical Society, Providence, RI, 2002

work page 2002
[33]

Mirkovi ´c and K

I. Mirkovi ´c and K. Vilonen, Geometric Langlands duality and representations of algebr aic groups over commutative rings , Ann. of Math. (2) 166 (2007), 95-143

work page 2007
[34]

Peterson, Quantum cohomology of G/P , Lecture course at MIT, Spring 1997

D. Peterson, Quantum cohomology of G/P , Lecture course at MIT, Spring 1997. Notes type- set by Arun Ram and Gil Azaria. Available at Lectures 1-5; 6-10; 11-15; 16-18

work page 1997
[35]

Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial ﬂag mani- folds, J

K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial ﬂag mani- folds, J. Amer. Math. Soc. 16 (2003), no. 2, 363-392

work page 2003
[36]

Rietsch, A mirror symmetric construction of qH ∗ T (G/P )(q), Adv

K. Rietsch, A mirror symmetric construction of qH ∗ T (G/P )(q), Adv. Math. 217 (2008), no. 6, 2401-2442

work page 2008
[37]

Rietsch, A mirror symmetric solution to the quantum Toda lattice , Comm

K. Rietsch, A mirror symmetric solution to the quantum Toda lattice , Comm. Math. Phys. 309 (2012), no. 1, 23-49

work page 2012
[38]

Y un and X

Z. Y un and X. Zhu, Integral homology of loop groups via Langlands dual groups , Represent. Theory 15 (2011), 347-369. 48 CHI HONG CHOW MAX PLANCK INSTITUTE FOR MATHEMATICS , 53111 B ONN , G ERMANY Email address: chow@mpim-bonn.mpg.de

work page 2011

[1] [1]

Baumann, J

P . Baumann, J. Kamnitzer and A. Knuston, The Mirkovi´ c-Vilonen basis and Duistermaat- Heckman measures. With an appendix by A. Dranowski, J. Kamni tzer and C. Morton- Ferguson, Acta Math. 227 (2021), no. 1, 1-101

work page 2021

[2] [2]

Berenstein and D

A. Berenstein and D. Kazhdan, Geometric and unipotent crystals , Geom. Funct. Anal., Spe- cial V olume, Part I (2000), 188-236

work page 2000

[3] [3]

Berenstein and D

A. Berenstein and D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicry s- tals to crystal bases , in: Quantum groups, Contemp. Math. 433, American Mathematical Society, Providence, RI, 2007, 13-88

work page 2007

[4] [4]

Berenstein and A

A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totall y positive varieties, Invent. Math. 143 (2001), no. 1, 77-128

work page 2001

[5] [5]

Berenstein and A

A. Berenstein and A. Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128-166

work page 1997

[6] [6]

Cheong and C

D. Cheong and C. Li, On the Conjecture O of GGI forG/P , Adv. Math. 306 (2017), 704-721

work page 2017

[7] [7]

Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion

R. Chhaibi, Littelmann path model for geometric crystals, Whittaker fu nctions on Lie groups and Brownian motion, Ph.D. Thesis, available at https://arxiv.org/abs/1302.0902. 46 CHI HONG CHOW

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

Chow, TheDℏ-module mirror conjecture for ﬂag varieties , Preprint (2023), available at https://arxiv.org/abs/2311.15523

C.H. Chow, TheDℏ-module mirror conjecture for ﬂag varieties , Preprint (2023), available at https://arxiv.org/abs/2311.15523

work page arXiv 2023

[9] [9]

Cox and S

D. Cox and S. Katz, Mirror symmetry and algebraic geometr y, in: Mathematical surveys and monographs, 68, American Mathematical Society, Providence, RI, 1999

work page 1999

[10] [10]

Galkin, J

S. Galkin, J. Hu, H. Iritani, H. Ke, C. Li and Z. Su, Revisiting Gamma conjecture I: coun- terexamples and modiﬁcations, Preprint (2024), available at https://arxiv.org/abs/2405.16979

work page arXiv 2024

[11] [11]

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

[12] [12]

Galkin and H

S. Galkin and H. Iritani, Gamma conjecture via mirror symmetry , in: Primitive forms and related subjects- Kavli IPMU 2014, Adv. Stud. Pure Math., 83, Math. Soc. Japan, 2019, 55- 115

work page 2014

[13] [13]

Gerasimov, S

A. Gerasimov, S. Kharchev, D. Lebedev and S. Oblezin, On a Gauss-Givental representation of quantum Toda chain wave function , Int. Math. Res. Not. IMRN, Art. ID 96489, 1-23

work page

[14] [14]

Gerasimov, D

A. Gerasimov, D. Lebedev and S. Oblezin, Baxter operator and Archimedean Hecke algebra , Comm. Math. Phys. 284 (2008), no. 3, 867-896

work page 2008

[15] [15]

Gerasimov, D

A. Gerasimov, D. Lebedev and S. Oblezin, New integral representations of Whittaker func- tions for classical Lie groups , Russ. Math. Surv. 67 (2012), no. 1, 1-92

work page 2012

[16] [16]

Givental, Stationary phase integrals, quantum Toda lattices, ﬂag man ifolds and the mir- ror conjecture, in: Topics in singularity theory: V

A. Givental, Stationary phase integrals, quantum Toda lattices, ﬂag man ifolds and the mir- ror conjecture, in: Topics in singularity theory: V . I. Arnold’s 60th anniv ersary collection, American Mathematical Society Translations: Series 2, 180, American Mathematical Soci- ety, Providence, RI, 1997, 103-115

work page 1997

[17] [17]

Golyshev and D

V . Golyshev and D. Zagier, Proof of the gamma conjecture for Fano 3-folds of Picard rank 1, Izv. Ross. Akad. Nauk Ser. Mat., 80 (2016), no. 1, 27-54

work page 2016

[18] [18]

Gross, Tropical geometry and mirror symmetry, in: CB MS regional conference series in Mathematics, 114, American Mathematical Society, Providence, RI, 2011

M. Gross, Tropical geometry and mirror symmetry, in: CB MS regional conference series in Mathematics, 114, American Mathematical Society, Providence, RI, 2011

work page 2011

[19] [19]

J. Hu, H. Ke, C. Li and T. Y ang, Gamma conjecture I for del Pezzo surfaces , Adv. Math. 386 (2021)

work page 2021

[20] [20]

Hugtenburg, On the quantum differential equations for a family of non-Kä hler monotone symplectic manifolds, Preprint (2024), available at https://arxiv.org/abs/2402.10867

K. Hugtenburg, On the quantum differential equations for a family of non-Kä hler monotone symplectic manifolds, Preprint (2024), available at https://arxiv.org/abs/2402.10867

work page arXiv 2024

[21] [21]

Iritani, An integral structure in quantum cohomology and mirror symm etry for toric orb- ifolds, Adv

H. Iritani, An integral structure in quantum cohomology and mirror symm etry for toric orb- ifolds, Adv. Math. 222 (2009), no. 3, 1016-1079

work page 2009

[22] [22]

Kashiwara, The crystal base and Littelmann’s reﬁned Demazure characte r formula, Duke Math

M. Kashiwara, The crystal base and Littelmann’s reﬁned Demazure characte r formula, Duke Math. J. 71 (1993), no. 3, 839-858. GAMMA CONJECTURE I FOR FLAG V ARIETIES 47

work page 1993

[23] [23]

Kim, Quantum cohomology of ﬂag manifolds G/B and quantum Toda lattices , Ann

B. Kim, Quantum cohomology of ﬂag manifolds G/B and quantum Toda lattices , Ann. of Math. (2) 149 (1999), no. 1, 129-148

work page 1999

[24] [24]

Knutson, T

A. Knutson, T. Lam and D.E. Speyer, Projections of Richardson varieties , J. Reine Angew. Math. 687 (2014), 133-157

work page 2014

[25] [25]

Kumar and M

S. Kumar and M. Nori, Positivity of the cup product in cohomology of ﬂag varieties associated to Kac-Moody groups, Int. Math. Res. Not. IMRN 1998 (1998), no. 14, 757-763

work page 1998

[26] [26]

Lam, Whittaker functions, geometric crystals, and quantum Schubert calculus, in: Schubert calculus - Osaka 2012, Adv

T. Lam, Whittaker functions, geometric crystals, and quantum Schubert calculus, in: Schubert calculus - Osaka 2012, Adv. Stud. Pure Math., 71, Math. Soc. Japan, 2016, 211-250

work page 2012

[27] [27]

Lam and K

T. Lam and K. Rietsch, Total positivity, Schubert positivity, and geometric Sata ke, J. Algebra 460 (2016), 284-319

work page 2016

[28] [28]

Lam and M

T. Lam and M. Shimozono, Quantum cohomology ofG/P and homology of afﬁne Grassman- nian, Acta Math. 204 (2010), no. 1, 49-90

work page 2010

[29] [29]

Lam and N

T. Lam and N. Templier, The mirror conjecture for minuscule ﬂag varieties , Duke Math. J. 173 (2024), no. 1, 75-175

work page 2024

[30] [30]

C. Li, K. Rietsch, M. Y ang and C. Zhang, A Plücker coordinate mirror for partial ﬂag varieties and quantum Schubert calculus , Preprint (2024), available at https://arxiv.org/abs/2401.15640

work page arXiv 2024

[31] [31]

Lusztig, Total positivity in reductive groups , in: Lie Theory and Geometry: In Honor of Bertram Kostant, Progr

G. Lusztig, Total positivity in reductive groups , in: Lie Theory and Geometry: In Honor of Bertram Kostant, Progr. in Math. 123, Birkhäuser, 1994, 531-568

work page 1994

[32] [32]

Manin, Frobenius manifolds, quantum cohomology and moduli spaces, in: Amer

Y . Manin, Frobenius manifolds, quantum cohomology and moduli spaces, in: Amer. Math. Soc. Colloq. Publ. 47, American Mathematical Society, Providence, RI, 2002

work page 2002

[33] [33]

Mirkovi ´c and K

I. Mirkovi ´c and K. Vilonen, Geometric Langlands duality and representations of algebr aic groups over commutative rings , Ann. of Math. (2) 166 (2007), 95-143

work page 2007

[34] [34]

Peterson, Quantum cohomology of G/P , Lecture course at MIT, Spring 1997

D. Peterson, Quantum cohomology of G/P , Lecture course at MIT, Spring 1997. Notes type- set by Arun Ram and Gil Azaria. Available at Lectures 1-5; 6-10; 11-15; 16-18

work page 1997

[35] [35]

Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial ﬂag mani- folds, J

K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial ﬂag mani- folds, J. Amer. Math. Soc. 16 (2003), no. 2, 363-392

work page 2003

[36] [36]

Rietsch, A mirror symmetric construction of qH ∗ T (G/P )(q), Adv

K. Rietsch, A mirror symmetric construction of qH ∗ T (G/P )(q), Adv. Math. 217 (2008), no. 6, 2401-2442

work page 2008

[37] [37]

Rietsch, A mirror symmetric solution to the quantum Toda lattice , Comm

K. Rietsch, A mirror symmetric solution to the quantum Toda lattice , Comm. Math. Phys. 309 (2012), no. 1, 23-49

work page 2012

[38] [38]

Y un and X

Z. Y un and X. Zhu, Integral homology of loop groups via Langlands dual groups , Represent. Theory 15 (2011), 347-369. 48 CHI HONG CHOW MAX PLANCK INSTITUTE FOR MATHEMATICS , 53111 B ONN , G ERMANY Email address: chow@mpim-bonn.mpg.de

work page 2011