Motivic Segre classes of Schubert cells and the connective formal group law

Raj Gandhi

REVIEW 1 major objections 1 minor 19 references

Motivic Segre classes of Schubert cells and the connective formal group law

T0 review · 1 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read The connective formal group law yields a β-deformation of motivic Segre classes of Schubert cells that specializes to known classes and admits explicit combinatorial descriptions for d=1.

desk verdict The paper defines a beta-deformation of motivic Segre classes via the connective formal group law, supplies d=1 lattice-model representatives and a puzzle formula proved with quantum-group intertwiners, and realizes them in a cobordism ring quotient after a GKM check. read the letter →

arxiv 2605.26556 v1 pith:ZTWQZDOS submitted 2026-05-26 math.CO math.ATmath.RT

Raj Gandhi This is my paper · ORCID

classification math.COmath.ATmath.RT

keywords motivicSegreclassesSchubertcellsconnectiveformalgrouplawβ-deformationKnutson-Taopuzzlesalgebraiccobordismflagvarietiesquantumgroups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines a one-parameter β-deformation of the motivic Segre classes of Schubert cells in d-step flag varieties using the connective formal group law. This deformation specializes to the original motivic classes when β equals 1 and to the Segre-Schwartz-MacPherson classes when β equals 0. In the d equals 1 case the deformed classes have explicit rational function representatives coming from a solvable lattice model. The structure constants in this deformed basis are given by a combinatorial formula proved with Knutson-Tao puzzles and intertwiners from a multi-parameter quantum group. The construction is realized as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety after verifying a GKM-type condition.

What carries the argument

The β-deformation of motivic Segre classes defined via the connective formal group law, with rational representatives from the solvable lattice model and structure constants from Knutson-Tao puzzles in the d=1 case.

What would settle it

A direct check for small d=1 cases in which the rational function representatives from the lattice model fail to match the expected specialization at β=1 would falsify the deformation construction.

Watch

Extended reading notes

Core claim

We use the connective formal group law to define a one-parameter (β-)deformation of the motivic Segre classes of Schubert cells in the d-step flag variety. This β-deformation specializes to the motivic Segre classes of Schubert cells when β=1 and to the Segre-Schwartz-MacPherson classes of Schubert cells when β=0. We define rational function representatives for the β-deformed classes in the d=1 case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the β-deformed basis in the d=1 case using Knutson-Tao puzzles. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type â2. We s

Load-bearing premise

The connective formal group law supplies a deformation of the motivic Segre classes that preserves the geometric and algebraic properties needed for the specializations at β=0 and β=1 to hold and for the GKM condition to be satisfiable in the cobordism ring.

Editorial extensions

If this is right

The β-deformed classes specialize to the motivic Segre classes when β=1.
The β-deformed classes specialize to the Segre-Schwartz-MacPherson classes when β=0.
The β-deformed classes in the d=1 case have rational function representatives from the solvable lattice model.
The structure constants of the β-deformed basis are given by the combinatorial puzzle formula.
The β-deformations correspond to quotients of canonical elements in the equivariant algebraic cobordism ring that satisfy a GKM condition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The lattice model solvability could be checked in higher d to test whether explicit representatives extend beyond the one-step case.
The quantum group intertwiners used for the puzzle proof may connect the construction to other representation-theoretic approaches in Schubert calculus.
The GKM verification step offers a method that might apply to deformations of similar classes on other homogeneous spaces.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The key thing to know is that this paper defines a beta-deformation of motivic Segre classes of Schubert cells using the connective formal group law. The deformation specializes to the usual motivic Segre classes at beta=1 and to Segre-Schwartz-MacPherson classes at beta=0. For the d=1 case it gives explicit rational function representatives from a solvable lattice model and proves a combinatorial formula for the structure constants via Knutson-Tao puzzles and intertwiners for the multi-parameter quantum group of type a2 hat. It also shows the classes arise as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle after verifying a GKM-type condition.

What is new is the specific use of the connective formal group law to produce this interpolation together with the d=1 combinatorial tools and the cobordism-ring interpretation. The paper does well in spelling out concrete representatives for d=1 and in connecting the structure constants to existing puzzle combinatorics and quantum-group methods.

The main place to check is whether the deformation preserves the geometric and algebraic properties needed for the specializations to hold and for the GKM condition to be satisfiable in the cobordism ring. The abstract states that the canonical elements satisfy the GKM-type condition, so that verification is the load-bearing step. If it is carried out correctly and without hidden choices, the rest follows. No other internal gaps are visible from the claims.

This is narrow work aimed at people working on motivic cohomology, Schubert calculus, or equivariant cobordism. A reader in that subfield would get value from the d=1 formulas and the deformation construction. It deserves a serious referee because the results are specific and the methods are standard enough to evaluate in detail. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript defines a one-parameter β-deformation of the motivic Segre classes of Schubert cells in the d-step flag variety via the connective formal group law. This deformation is asserted to specialize to the motivic Segre classes at β=1 and to the Segre-Schwartz-MacPherson classes at β=0. For the d=1 case, rational-function representatives are constructed via a solvable lattice model, and a combinatorial formula for the structure constants in the deformed basis is proved using Knutson-Tao puzzles together with intertwiners for the multi-parameter quantum group of type â2. The deformed classes are realized as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety, after verifying a GKM-type condition on those elements.

Significance. If the compatibility of the deformation with the required geometric and algebraic properties holds, the work supplies an explicit bridge between motivic and classical Segre classes together with combinatorial tools (lattice models, puzzle formulas) and a cobordism-ring interpretation. The explicit use of solvable lattice models and the proof of the puzzle formula via quantum-group intertwiners constitute concrete, verifiable contributions for the d=1 case.

major comments (1)

[construction of the β-deformed classes and the GKM verification paragraph] The central construction (abstract and opening sections) defines the β-deformed classes directly from the connective formal group law and asserts that the resulting objects specialize correctly at β=0,1 while satisfying the GKM condition needed for the cobordism interpretation. Because this compatibility is invoked both for the specializations and for the final realization as quotients of canonical elements, an explicit verification that the lattice-model representatives and the GKM condition remain valid for generic β (rather than only at the endpoints) is required; without it, the combinatorial formula and the cobordism claim lose their claimed geometric meaning.

minor comments (1)

[d=1 combinatorial section] Notation for the multi-parameter quantum group of type â2 and the precise definition of the lattice model should be introduced with a short self-contained paragraph or diagram to aid readers unfamiliar with the â2 intertwiners.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the central construction. We address the point below and will revise the manuscript to incorporate an explicit verification as requested.

read point-by-point responses

Referee: The central construction (abstract and opening sections) defines the β-deformed classes directly from the connective formal group law and asserts that the resulting objects specialize correctly at β=0,1 while satisfying the GKM condition needed for the cobordism interpretation. Because this compatibility is invoked both for the specializations and for the final realization as quotients of canonical elements, an explicit verification that the lattice-model representatives and the GKM condition remain valid for generic β (rather than only at the endpoints) is required; without it, the combinatorial formula and the cobordism claim lose their claimed geometric meaning.

Authors: We agree that an explicit verification for generic β is necessary to fully support the geometric claims. The β-deformed classes are constructed algebraically via the connective formal group law over the ring ℤ[β], and the lattice-model representatives for d=1 are defined with weights that are polynomials in β. The GKM-type condition on the canonical elements in the quotient of the equivariant algebraic cobordism ring can therefore be verified by direct (albeit tedious) computation that holds identically as an identity in β, after which the specializations at β=0 and β=1 are obtained by substitution. The quantum-group intertwiners used for the puzzle formula are likewise defined over the multi-parameter ring that includes β. To make this verification fully explicit rather than implicit in the algebraic setup, we will add a dedicated subsection (new Section 4.3 or equivalent) that carries out the GKM check for a generic β and records the resulting identities. This revision will be included in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: deformation defined from standard object with independent combinatorial and geometric proofs

full rationale

The paper defines the β-deformation of motivic Segre classes directly from the connective formal group law (a pre-existing standard construction) and then derives rational representatives via a lattice model, proves structure constants via Knutson-Tao puzzles and quantum-group intertwiners, and realizes the classes in the cobordism ring after an explicit GKM verification. These steps are presented as independent derivations rather than reductions to fitted inputs or self-citations. No equation or claim reduces a result to its own definition by construction.

Assumptions & free parameters 1 free parameters · 2 assumptions · 1 invented entities

The central construction rests on the connective formal group law as a background object from algebraic topology and on the existence of canonical elements in the equivariant algebraic cobordism ring; the paper introduces the β-deformed classes as a new object.

free parameters (1)

β
Deformation parameter drawn from the connective formal group law; controls the interpolation between motivic and classical classes.

assumptions (2)

domain assumption The connective formal group law can be used to deform motivic Segre classes while preserving the required specialization properties at β=0 and β=1.
Invoked at the start of the definition of the β-deformation.
domain assumption Canonical elements exist in the quotient of the equivariant algebraic cobordism ring and satisfy a GKM-type condition.
Used to interpret the β-deformations as quotients.

invented entities (1)

β-deformed motivic Segre classes
purpose: To provide a one-parameter family interpolating between motivic Segre classes and Segre-Schwartz-MacPherson classes.
Newly defined object whose properties are proved in the paper.

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Cite this review

Pith. "Pith review of Motivic Segre classes of Schubert cells and the connective formal group law." pith.science (2026). https://pith.science/paper/ZTWQZDOS

@misc{pith2026260526556,
  author       = {Pith},
  title        = {Pith review of: Motivic Segre classes of Schubert cells and the connective formal group law},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTWQZDOS}},
  note         = {Machine review of arXiv:2605.26556}
}

abstract

We use the connective formal group law to define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $\beta$-deformation specializes to the motivic Segre classes of Schubert cells when $\beta=1$ and to the Segre-Schwartz-MacPherson classes of Schubert cells when $\beta=0$. We define rational function representatives for the $\beta$-deformed classes in the $d=1$ case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the $\beta$-deformed basis in the $d=1$ case using Knutson-Tao puzzles. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type $\widehat{a}_2$. We show that our $\beta$-deformations can be viewed as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Mihalcea

[AM16] Paolo Aluffi and Leonardo C. Mihalcea. Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds.Compos. Math., 152(12):2603–2625, 2016.doi:10.1112/S0010437X16007685. [AMSS23] Paolo Aluffi, Leonardo C. Mihalcea, J ¨org Sch ¨urmann, and Changjian Su. Shadows of characteristic cycles, Verma mod- ules, and positivity of Chern-Schwartz-Mac...

work page doi:10.1112/s0010437x16007685 2016

[2] [2]

[AST91] Michael Artin, William Schelter, and John Tate

doi:10.1215/00127094-2022-0101. [AST91] Michael Artin, William Schelter, and John Tate. Quantum deformations ofGL n.Comm. Pure Appl. Math., 44(8-9):879–895, 1991.doi:10.1002/cpa.3160440804. [BB73] A. Białynicki-Birula. Some theorems on actions of algebraic groups.Ann. of Math. (2), 98:480–497, 1973.doi:10.2307/ 1970915. [BE90] Paul Bressler and Sam Evens....

work page doi:10.1215/00127094-2022-0101 2022

[3] [3]

[Bri97] M

doi:10.2307/2154142. [Bri97] M. Brion. Equivariant Chow groups for torus actions.Transform. Groups, 2(3):225–267, 1997.doi:10.1007/BF01234659. [BSY10] Jean-Paul Brasselet, J ¨org Sch¨urmann, and Shoji Yokura. Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal., 2(1):1–55, 2010.doi:10.1142/S1793525310000239. [Cai08] Shuang Cai...

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and Zainoulline, K

https://arxiv.org/abs/1312.0019.doi:10.1007/s00229-018-1058-4. [Dem73] Michel Demazure. Invariants sym ´etriques entiers des groupes de Weyl et torsion.Invent. Math., 21:287–301, 1973.doi: 10.1007/BF01418790. [Dem74] Michel Demazure. Une nouvelle formule des caract `eres.Bull. Sci. Math. (2), 98(3):163–172,

work page Pith review arXiv doi:10.1007/s00229-018-1058-4 1973

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[Des] Dinesh Deshpande. Algebraic cobordism of classifying spaces. https://arxiv.org/abs/0907.4437v1. [DL14] Shouxin Dai and Marc Levine. Connective algebraicK-theory.J. K-Theory, 13(1):9–56, 2014.doi:10.1017/ is013012007jkt249. 52Raj Gandhi [EG98] Dan Edidin and William Graham. Equivariant intersection theory.Invent. Math., 131(3):595–634, 1998.doi:10.10...

work page Pith review arXiv 2014

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1995 , PAGES =

With a view toward algebraic geometry.doi:10.1007/978-1-4612-5350-1. [FKsd] Sergey Fomin and Anatol N. Kirillov. Grothendieck polynomials and the Yang-Baxter equation. InFormal power series and algebraic combinatorics/S´ eries formelles et combinatoire alg´ ebrique, pages 183–189. DIMACS, Piscataway, NJ, sd. [Fr¨o68] A. Fr ¨ohlich.Formal groups. Lecture N...

work page doi:10.1007/978-1-4612-5350-1

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Feh ´er, Rich ´ard Rim ´anyi, and Andrzej Weber

[FRW21] L ´aszl´o M. Feh ´er, Rich ´ard Rim ´anyi, and Andrzej Weber. Motivic Chern classes and K-theoretic stable envelopes.Proc. Lond. Math. Soc. (3), 122(1):153–189, 2021.doi:10.1112/plms.12374. [Gan] Raj Gandhi. Sagemath scripts. https://github.com/RajGandhi97/Motivic-Segre-classes-of-Schubert-cells-and-the- connective-formal-group-law. Accessed 13 Ja...

work page doi:10.1112/plms.12374 2021