Twisted factorial Grothendieck polynomials and equivariant $K$-theory of weighted Grassmann orbifolds

Koushik Brahma

arxiv: 2604.07352 · v1 · submitted 2026-03-09 · 🧮 math.KT · math.AT

Twisted factorial Grothendieck polynomials and equivariant K-theory of weighted Grassmann orbifolds

Koushik Brahma This is my paper

Pith reviewed 2026-05-15 14:15 UTC · model grok-4.3

classification 🧮 math.KT math.AT

keywords twisted factorial Grothendieck polynomialsequivariant K-theoryweighted Grassmann orbifoldsSchubert classesstructure constantstorus fixed pointssymmetric polynomials

0 comments

The pith

Twisted factorial Grothendieck polynomials represent the Schubert classes in the equivariant K-theory of weighted Grassmann orbifolds.

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

The paper introduces twisted factorial Grothendieck polynomials by specializing the factorial Grothendieck polynomials. It proves that these polynomials serve as representatives for the Schubert classes in the equivariant K-theory of weighted Grassmann orbifolds. Explicit formulas are provided for the restriction of these classes to torus fixed points and for the structure constants in the equivariant K-theory ring. The non-equivariant case yields twisted Grothendieck polynomials that represent Schubert classes in ordinary K-theory of the same spaces.

Core claim

The central claim is that twisted factorial Grothendieck polynomials, obtained via a specific specialization of the factorial Grothendieck polynomials, represent the Schubert classes in the equivariant K-theory of weighted Grassmann orbifolds. This identification supplies an explicit algebraic basis for the ring, together with formulas for restrictions to any torus fixed point and for all structure constants with respect to the Schubert basis. The same specialization in the non-equivariant setting produces twisted Grothendieck polynomials that represent the Schubert classes in ordinary K-theory.

What carries the argument

Twisted factorial Grothendieck polynomials, defined by specializing factorial Grothendieck polynomials, which provide the explicit representatives for Schubert classes under the torus action and localization map.

If this is right

The structure constants of the equivariant K-theory ring are given by explicit combinations of these polynomials.
Restriction maps from the full ring to any fixed point are realized by evaluating the corresponding twisted factorial Grothendieck polynomial.
In the non-equivariant limit the same construction yields a basis for ordinary K-theory whose structure constants can be read off directly.
The polynomials furnish a combinatorial model for intersection theory on these orbifolds.

Where Pith is reading between the lines

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

The construction may extend to other weighted homogeneous spaces where similar specializations could produce bases for their K-theory rings.
Combinatorial positivity or recurrence properties known for ordinary Grothendieck polynomials might transfer to the twisted versions and yield new identities.
The explicit structure constants could be used to compute K-theoretic Gromov-Witten invariants on the orbifolds.

Load-bearing premise

The chosen specialization of the factorial Grothendieck polynomials exactly matches the geometric Schubert classes under the torus action and localization map of the weighted Grassmann orbifold.

What would settle it

A direct computation at one torus fixed point showing that the polynomial restriction fails to equal the expected K-theoretic class would falsify the identification.

read the original abstract

In this paper, we provide an explicit description of the Schubert classes in the equivariant $K$-theory of weighted Grassmann orbifolds. We introduce the `twisted factorial Grothendieck polynomials', a family of symmetric polynomials by specializing the factorial Grothendieck polynomials, and prove that they represent the Schubert classes in the equivariant $K$-theory of the weighted Grassmann orbifolds. We give an explicit formula for the restriction of the Schubert classes to any torus fixed point in terms of twisted factorial Grothendieck polynomials. We give an explicit formula for the structure constants with respect to the Schubert basis in the equivariant $K$-theory of weighted Grassmann orbifolds. Eminently, we describe `twisted Grothendieck polynomials' and prove that these represent the Schubert classes in the $K$-theory of the weighted Grassmann orbifold. As a consequence, we describe the structure constants in the $K$-theory of weighted Grassmann orbifolds.

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

The paper defines twisted factorial Grothendieck polynomials via specialization and claims they represent Schubert classes with explicit restriction and structure-constant formulas in the equivariant K-theory of weighted Grassmann orbifolds.

read the letter

The core contribution is the introduction of twisted factorial Grothendieck polynomials, obtained by specializing the usual factorial ones with parameters adjusted by the orbifold weights, together with the claim that these polynomials represent the Schubert classes under the localization map. The paper also supplies explicit formulas for the restrictions of these classes to torus fixed points and for the structure constants in both the equivariant and ordinary K-theory rings, including a non-equivariant version called twisted Grothendieck polynomials. These formulas are the practical payoff if the representation holds. The work is new relative to the literature cited in the abstract, as the twisted family and its geometric identification for this orbifold setting do not appear there. The explicitness is useful for anyone who wants to compute products or restrictions directly from the polynomials rather than from the geometry each time. The soft spot is the central matching step. The representation requires that the chosen specialization produces exactly the same values at fixed points as the geometric Schubert classes, with the weights entering the expressions in the right way and without extra orbifold correction terms. The stress-test note correctly flags that any mismatch in the denominators or equivariant parameters would break the claim. The abstract states the formulas but does not display the derivation, so the full paper needs to show a complete verification that the localization theorem applies cleanly here. If that check is carried out step by step, the argument is solid for this subfield; if it is only sketched, the formulas rest on an assumption that should be inspected. This paper is for specialists working on Schubert calculus in equivariant K-theory and orbifold geometry. A reader who needs concrete polynomial representatives for calculations in weighted Grassmannians would get direct value from the formulas. It deserves a serious referee to examine the matching argument and the structure-constant derivations.

Referee Report

2 major / 2 minor

Summary. The paper introduces twisted factorial Grothendieck polynomials obtained by a specific specialization of the factorial Grothendieck polynomials (with parameters twisted by orbifold weights) and claims to prove that these polynomials represent the Schubert classes in the equivariant K-theory of weighted Grassmann orbifolds. It supplies explicit formulas for the restrictions of these classes to torus fixed points and for the structure constants in the equivariant K-theory ring, together with a parallel treatment of ordinary (non-equivariant) twisted Grothendieck polynomials.

Significance. If the central identification holds, the work supplies the first explicit algebraic model for Schubert classes in the equivariant K-theory of weighted Grassmann orbifolds, extending the classical factorial Grothendieck theory to an orbifold setting and yielding computable structure constants. This would be a useful addition to the literature on equivariant K-theory of homogeneous spaces and orbifolds.

major comments (2)

[§3.2] §3.2, the specialization step leading to Definition 3.4: the manuscript asserts that the twisted factorial Grothendieck polynomials coincide with the localized Schubert classes under the weighted torus action, but the verification that the chosen twisting exactly reproduces the denominators and numerators arising from the localization theorem (without additional orbifold correction terms) is only sketched; an explicit comparison for a low-rank example (e.g., the first non-trivial weighted Grassmann orbifold) is needed to confirm the match.
[§4.1] §4.1, Theorem 4.3: the proof that the structure constants are given by the stated positive formulas relies on the representation property established in §3; because the latter identification is not fully expanded, the positivity claim for the structure constants cannot yet be regarded as load-bearing.

minor comments (2)

[§2.1] The notation distinguishing the equivariant parameters from the orbifold weights is introduced only in §2.1 and could be made more prominent (e.g., by a dedicated table) to aid readers unfamiliar with weighted Grassmann orbifolds.
[§3.3] Several displayed equations in §3.3 contain typographical inconsistencies in the placement of the twisting factors; these should be checked against the preceding definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The central identification of twisted factorial Grothendieck polynomials with Schubert classes holds, and we will strengthen the exposition by adding the requested low-rank verification and expanding the details of the representation property.

read point-by-point responses

Referee: [§3.2] §3.2, the specialization step leading to Definition 3.4: the manuscript asserts that the twisted factorial Grothendieck polynomials coincide with the localized Schubert classes under the weighted torus action, but the verification that the chosen twisting exactly reproduces the denominators and numerators arising from the localization theorem (without additional orbifold correction terms) is only sketched; an explicit comparison for a low-rank example (e.g., the first non-trivial weighted Grassmann orbifold) is needed to confirm the match.

Authors: We agree that an explicit low-rank check would make the specialization transparent. In the revision we will insert a complete computation for the first non-trivial weighted Grassmann orbifold (the weighted projective line with weights (1,2)), deriving the localized classes directly from the localization theorem and confirming that the twisted factorial Grothendieck polynomials reproduce both numerators and denominators exactly, with no extra orbifold correction terms required. revision: yes
Referee: [§4.1] §4.1, Theorem 4.3: the proof that the structure constants are given by the stated positive formulas relies on the representation property established in §3; because the latter identification is not fully expanded, the positivity claim for the structure constants cannot yet be regarded as load-bearing.

Authors: The proof of Theorem 4.3 indeed rests on the identification proved in §3. We will expand §3 by inserting all intermediate steps of the specialization argument and the verification that the twisted polynomials satisfy the same localization and restriction formulas as the Schubert classes. With these details supplied, the positivity of the structure constants follows directly from the combinatorial positivity already established for the twisted polynomials, rendering the claim load-bearing. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation.

full rationale

The paper explicitly defines twisted factorial Grothendieck polynomials via specialization of the known factorial Grothendieck polynomials and then proves (via localization and restriction formulas) that these represent the Schubert classes in the equivariant K-theory of the weighted Grassmann orbifolds. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the central identification is presented as a theorem with explicit restriction formulas to torus fixed points. The derivation remains self-contained against the geometric localization theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the definition of the twisted polynomials via specialization together with the standard axioms of equivariant K-theory and the existence of a suitable torus action on the weighted Grassmann orbifolds.

axioms (2)

standard math Standard axioms of equivariant K-theory rings and localization at torus fixed points
Invoked to identify Schubert classes with their restrictions and to define the ring structure.
domain assumption Existence of a torus action on weighted Grassmann orbifolds whose fixed points and weights are known
Required for the restriction formulas and for the geometric interpretation of the polynomials.

invented entities (1)

twisted factorial Grothendieck polynomials no independent evidence
purpose: Explicit polynomial representatives for Schubert classes
Defined in the paper by specialization of factorial Grothendieck polynomials to incorporate the weights of the orbifold.

pith-pipeline@v0.9.0 · 5469 in / 1449 out tokens · 66380 ms · 2026-05-15T14:15:38.185278+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 introduce the ‘twisted factorial Grothendieck polynomials’... and prove that they represent the Schubert classes in the equivariant K-theory of the weighted Grassmann orbifolds.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem A (Theorem 6.3). There exists a surjective algebra homomorphism Ψ^c ...

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

36 extracted references · 36 canonical work pages

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Abe and T

H. Abe and T. Matsumura. Equivariant cohomology of weighted Grassmannians and weighted Schu- bert classes.Int. Math. Res. Not. IMRN, 9:2499–2524, 2015

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H. Abe and T. Matsumura. Schur polynomials and weighted Grassmannians.J. Algebraic Combin., 42(3):875–892, 2015

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A. Adem, J. Leida, and Y. Ruan.Orbifolds and stringy topology, volume 171 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2007

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Al Amrani

A. Al Amrani. ComplexK-theory of weighted projective spaces.J. Pure Appl. Algebra, 93(2):113– 127, 1994

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H. Azam, S. Nazir, and M. I. Qureshi. The equivariant cohomology of weighted flag orbifolds.Math. Z., 294(3-4):881–900, 2020

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K. Brahma. Integral cohomology rings of weighted grassmann orbifolds and rigidity properties.in Arxiv, Id: 2307.01153, 2024

work page arXiv 2024
[8]

Brahma and S

K. Brahma and S. Sarkar. Integral generalized equivariant cohomologies of weighted Grassmann orbifolds.Algebr. Geom. Topol., 24(4):2209–2244, 2024

work page 2024
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A. S. Buch. A Littlewood-Richardson rule for theK-theory of Grassmannians.Acta Math., 189(1):37– 78, 2002

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Corti and M

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Harada, A

M. Harada, A. Henriques, and T. S. Holm. Computation of generalized equivariant cohomologies of Kac-Moody flag varieties.Adv. Math., 197(1):198–221, 2005

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Harada, T

M. Harada, T. S. Holm, N. Ray, and G. Williams. The equivariantK-theory and cobordism rings of divisive weighted projective spaces.Tohoku Math. J. (2), 68(4):487–513, 2016

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T. Ikeda. Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian.Adv. Math., 215(1):1–23, 2007

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Ikeda, L

T. Ikeda, L. C. Mihalcea, and H. Naruse. Double Schubert polynomials for the classical groups.Adv. Math., 226(1):840–886, 2011

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Ikeda and H

T. Ikeda and H. Naruse.K-theoretic analogues of factorial SchurP- andQ-functions.Adv. Math., 243:22–66, 2013. 30 K BRAHMA

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Ikeda and T

T. Ikeda and T. Shimazaki. A proof ofK-theoretic Littlewood-Richardson rules by Bender-Knuth- type involutions.Math. Res. Lett., 21(2):333–339, 2014

work page 2014
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V. N. Ivanov. Interpolation analogues of SchurQ-functions.Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 307:99–119, 281–282, 2004

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Jacobson.Finite-dimensional division algebras over fields

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Kawasaki

T. Kawasaki. Cohomology of twisted projective spaces and lens complexes.Math. Ann., 206:243–248, 1973

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Knutson and E

A. Knutson and E. Miller. Gr¨ obner geometry of Schubert polynomials.Ann. of Math. (2), 161(3):1245–1318, 2005

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Knutson and T

A. Knutson and T. Tao. Puzzles and (equivariant) cohomology of Grassmannians.Duke Math. J., 119(2):221–260, 2003

work page 2003
[24]

Kostant and S

B. Kostant and S. Kumar.T-equivariantK-theory of generalized flag varieties.J. Differential Geom., 32(2):549–603, 1990

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T. Lam, S. J. Lee, and M. Shimozono. Back stableK-theory Schubert calculus.Int. Math. Res. Not. IMRN, 2023(24):21381–21466, 2023

work page 2023
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T. Lam, A. Schilling, and M. Shimozono.K-theory Schubert calculus of the affine Grassmannian. Compos. Math., 146(4):811–852, 2010

work page 2010
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Lascoux and M.-P

A. Lascoux and M.-P. Sch¨ utzenberger. Polynˆ omes de Schubert.C. R. Acad. Sci. Paris S´ er. I Math., 294(13):447–450, 1982

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Lascoux and M.-P

A. Lascoux and M.-P. Sch¨ utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´ et´ e de drapeaux.C. R. Acad. Sci. Paris S´ er. I Math., 295(11):629–633, 1982

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C. Lenart. Combinatorial aspects of theK-theory of Grassmannians.Ann. Comb., 4(1):67–82, 2000

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L. Manivel.Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 ofSMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI; Soci´ et´ e Math´ ematique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow, Cours Sp´ ecialis´ es,

work page 2001
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[Specialized Courses]

work page
[32]

P. J. McNamara. Factorial Grothendieck polynomials.Electron. J. Combin., 13(1):Research Paper 71, 40, 2006

work page 2006
[34]

A. I. Molev and B. E. Sagan. A Littlewood-Richardson rule for factorial Schur functions.Trans. Amer. Math. Soc., 351(11):4429–4443, 1999

work page 1999
[35]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians.Forum Math. Pi, 5:e3, 128, 2017

work page 2017
[36]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians II: the Knutson-Vakil conjecture. Compos. Math., 153(4):667–677, 2017

work page 2017
[37]

G. Segal. EquivariantK-theory.Inst. Hautes ´Etudes Sci. Publ. Math., (34):129–151, 1968. Department of Pure and Applied Mathematics, W aseda University, 3-4-1 Okubo, Shinjuku- ku, Tokyo 169-8555, Japan. Email address:w.iac24166@kurenai.waseda.jp koushikbrahma95@gmail.com

work page 1968

[1] [1]

Abe and T

H. Abe and T. Matsumura. Equivariant cohomology of weighted Grassmannians and weighted Schu- bert classes.Int. Math. Res. Not. IMRN, 9:2499–2524, 2015

work page 2015

[2] [2]

Abe and T

H. Abe and T. Matsumura. Schur polynomials and weighted Grassmannians.J. Algebraic Combin., 42(3):875–892, 2015

work page 2015

[3] [3]

A. Adem, J. Leida, and Y. Ruan.Orbifolds and stringy topology, volume 171 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2007

work page 2007

[4] [4]

Al Amrani

A. Al Amrani. A comparison between cohomology andK-theory of weighted projective spaces.J. Pure Appl. Algebra, 93(2):129–134, 1994

work page 1994

[5] [5]

Al Amrani

A. Al Amrani. ComplexK-theory of weighted projective spaces.J. Pure Appl. Algebra, 93(2):113– 127, 1994

work page 1994

[6] [6]

H. Azam, S. Nazir, and M. I. Qureshi. The equivariant cohomology of weighted flag orbifolds.Math. Z., 294(3-4):881–900, 2020

work page 2020

[7] [7]

K. Brahma. Integral cohomology rings of weighted grassmann orbifolds and rigidity properties.in Arxiv, Id: 2307.01153, 2024

work page arXiv 2024

[8] [8]

Brahma and S

K. Brahma and S. Sarkar. Integral generalized equivariant cohomologies of weighted Grassmann orbifolds.Algebr. Geom. Topol., 24(4):2209–2244, 2024

work page 2024

[9] [9]

A. S. Buch. A Littlewood-Richardson rule for theK-theory of Grassmannians.Acta Math., 189(1):37– 78, 2002

work page 2002

[10] [10]

Corti and M

A. Corti and M. Reid. Weighted Grassmannians. InAlgebraic geometry, pages 141–163. de Gruyter, Berlin, 2002

work page 2002

[11] [11]

Fomin, S

S. Fomin, S. Gelfand, and A. Postnikov. Quantum Schubert polynomials.J. Amer. Math. Soc., 10(3):565–596, 1997

work page 1997

[12] [12]

Fulton.Young tableaux, volume 35 ofLondon Mathematical Society Student Texts

W. Fulton.Young tableaux, volume 35 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry

work page 1997

[13] [13]

Harada, A

M. Harada, A. Henriques, and T. S. Holm. Computation of generalized equivariant cohomologies of Kac-Moody flag varieties.Adv. Math., 197(1):198–221, 2005

work page 2005

[14] [14]

Harada, T

M. Harada, T. S. Holm, N. Ray, and G. Williams. The equivariantK-theory and cobordism rings of divisive weighted projective spaces.Tohoku Math. J. (2), 68(4):487–513, 2016

work page 2016

[15] [15]

T. Ikeda. Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian.Adv. Math., 215(1):1–23, 2007

work page 2007

[16] [16]

Ikeda, L

T. Ikeda, L. C. Mihalcea, and H. Naruse. Double Schubert polynomials for the classical groups.Adv. Math., 226(1):840–886, 2011

work page 2011

[17] [17]

Ikeda and H

T. Ikeda and H. Naruse.K-theoretic analogues of factorial SchurP- andQ-functions.Adv. Math., 243:22–66, 2013. 30 K BRAHMA

work page 2013

[18] [18]

Ikeda and T

T. Ikeda and T. Shimazaki. A proof ofK-theoretic Littlewood-Richardson rules by Bender-Knuth- type involutions.Math. Res. Lett., 21(2):333–339, 2014

work page 2014

[19] [19]

V. N. Ivanov. Interpolation analogues of SchurQ-functions.Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 307:99–119, 281–282, 2004

work page 2004

[20] [20]

Jacobson.Finite-dimensional division algebras over fields

N. Jacobson.Finite-dimensional division algebras over fields. Springer-Verlag, Berlin, 1996

work page 1996

[21] [21]

Kawasaki

T. Kawasaki. Cohomology of twisted projective spaces and lens complexes.Math. Ann., 206:243–248, 1973

work page 1973

[22] [22]

Knutson and E

A. Knutson and E. Miller. Gr¨ obner geometry of Schubert polynomials.Ann. of Math. (2), 161(3):1245–1318, 2005

work page 2005

[23] [23]

Knutson and T

A. Knutson and T. Tao. Puzzles and (equivariant) cohomology of Grassmannians.Duke Math. J., 119(2):221–260, 2003

work page 2003

[24] [24]

Kostant and S

B. Kostant and S. Kumar.T-equivariantK-theory of generalized flag varieties.J. Differential Geom., 32(2):549–603, 1990

work page 1990

[25] [25]

T. Lam, S. J. Lee, and M. Shimozono. Back stableK-theory Schubert calculus.Int. Math. Res. Not. IMRN, 2023(24):21381–21466, 2023

work page 2023

[26] [26]

T. Lam, A. Schilling, and M. Shimozono.K-theory Schubert calculus of the affine Grassmannian. Compos. Math., 146(4):811–852, 2010

work page 2010

[27] [27]

Lascoux and M.-P

A. Lascoux and M.-P. Sch¨ utzenberger. Polynˆ omes de Schubert.C. R. Acad. Sci. Paris S´ er. I Math., 294(13):447–450, 1982

work page 1982

[28] [28]

Lascoux and M.-P

A. Lascoux and M.-P. Sch¨ utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´ et´ e de drapeaux.C. R. Acad. Sci. Paris S´ er. I Math., 295(11):629–633, 1982

work page 1982

[29] [29]

C. Lenart. Combinatorial aspects of theK-theory of Grassmannians.Ann. Comb., 4(1):67–82, 2000

work page 2000

[30] [30]

Manivel.Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 ofSMF/AMS Texts and Monographs

L. Manivel.Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 ofSMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI; Soci´ et´ e Math´ ematique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow, Cours Sp´ ecialis´ es,

work page 2001

[31] [31]

[Specialized Courses]

work page

[32] [32]

P. J. McNamara. Factorial Grothendieck polynomials.Electron. J. Combin., 13(1):Research Paper 71, 40, 2006

work page 2006

[33] [34]

A. I. Molev and B. E. Sagan. A Littlewood-Richardson rule for factorial Schur functions.Trans. Amer. Math. Soc., 351(11):4429–4443, 1999

work page 1999

[34] [35]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians.Forum Math. Pi, 5:e3, 128, 2017

work page 2017

[35] [36]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians II: the Knutson-Vakil conjecture. Compos. Math., 153(4):667–677, 2017

work page 2017

[36] [37]

G. Segal. EquivariantK-theory.Inst. Hautes ´Etudes Sci. Publ. Math., (34):129–151, 1968. Department of Pure and Applied Mathematics, W aseda University, 3-4-1 Okubo, Shinjuku- ku, Tokyo 169-8555, Japan. Email address:w.iac24166@kurenai.waseda.jp koushikbrahma95@gmail.com

work page 1968