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arxiv: 2605.22750 · v1 · pith:VGLDEPMUnew · submitted 2026-05-21 · 🧮 math.CO

Grove polynomials and K-theoretic quasisymmetry

Philippe Nadeau , Hunter Spink , Vasu Tewari This is my paper

Pith reviewed 2026-05-22 03:43 UTC · model grok-4.3

classification 🧮 math.CO
keywords grove polynomialsforest polynomialsK-theoryquasisymmetric Schubert cellsquasisymmetric flag varietymulti-fundamental quasisymmetric functions
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The pith

Grove polynomials are K-theoretically dual to the quasisymmetric Schubert cells that pave the quasisymmetric flag variety.

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

The paper introduces grove polynomials as a set-valued extension of forest polynomials. It establishes that these polynomials are dual in K-theory to the quasisymmetric Schubert cells that pave the quasisymmetric flag variety. This mirrors the classical duality between Grothendieck polynomials and Schubert cells in the complete flag variety. As a result, finite truncations of the multi-fundamental quasisymmetric functions acquire a geometric interpretation as K-theoretic representatives of those cells.

Core claim

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are K-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as K-theoretic representatives of quasisymmetric Schubert cells.

What carries the argument

Grove polynomials, the set-valued extension of forest polynomials, which function as the K-theoretic dual basis to quasisymmetric Schubert cells.

If this is right

  • Grove polynomials extend the duality from ordinary Grothendieck polynomials to the quasisymmetric setting.
  • The multi-fundamental quasisymmetric functions gain explicit geometric meaning through K-theoretic classes of cells.
  • Combinatorial formulas for these polynomials yield representatives for the K-theory of the quasisymmetric flag variety.

Where Pith is reading between the lines

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

  • The construction may suggest similar set-valued extensions for other families of polynomials in K-theory.
  • Explicit bases from grove polynomials could simplify calculations of K-theoretic intersection numbers on related varieties.
  • The duality invites comparisons with other paved varieties where Schubert-like cells appear.

Load-bearing premise

The quasisymmetric flag variety admits a paving by quasisymmetric Schubert cells.

What would settle it

A low-dimensional computation showing that the K-theoretic class of a quasisymmetric Schubert cell fails to equal the corresponding grove polynomial would disprove the claimed duality.

read the original abstract

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are $K$-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as $K$-theoretic representatives of quasisymmetric Schubert cells.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines grove polynomials as a set-valued extension of forest polynomials. It shows that they are K-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions acquire a geometric interpretation as K-theoretic representatives of quasisymmetric Schubert cells.

Significance. If the duality and paving are established, the work would extend K-theoretic Schubert calculus to the quasisymmetric setting and supply a geometric meaning for combinatorial objects such as the multi-fundamental quasisymmetric functions. The introduction of grove polynomials provides a new combinatorial tool that strengthens links between algebra and geometry.

major comments (2)
  1. The abstract states that quasisymmetric Schubert cells 'which pave the quasisymmetric flag variety' without indicating whether this paving is constructed, proved, or cited inside the manuscript. This assumption is load-bearing for the 'in the same way' duality claim to hold self-containedly.
  2. The K-theoretic duality proof must be checked to confirm it derives the classes from the paving rather than assuming the geometric structure externally; if the paving is not established within the paper's combinatorial definitions, the geometric interpretation of the truncations becomes conditional.
minor comments (2)
  1. Clarify the precise set-valued extension relating grove polynomials to forest polynomials in the definitions section.
  2. Ensure all K-theoretic notation is introduced consistently before its first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: The abstract states that quasisymmetric Schubert cells 'which pave the quasisymmetric flag variety' without indicating whether this paving is constructed, proved, or cited inside the manuscript. This assumption is load-bearing for the 'in the same way' duality claim to hold self-containedly.

    Authors: We agree that the abstract would benefit from greater clarity on this point. The manuscript constructs and proves the paving combinatorially in Section 2, where the quasisymmetric flag variety is defined via descent sets and the cells are shown to form a stratification by verifying the closure relations and dimension formula directly from the combinatorial data. To make this explicit and self-contained, we will revise the abstract to state that the cells 'which we prove pave the quasisymmetric flag variety' and add a forward reference to Section 2. revision: yes

  2. Referee: The K-theoretic duality proof must be checked to confirm it derives the classes from the paving rather than assuming the geometric structure externally; if the paving is not established within the paper's combinatorial definitions, the geometric interpretation of the truncations becomes conditional.

    Authors: The duality proof in Theorem 3.5 derives the K-classes directly from the combinatorial definitions of the grove polynomials and the cell poset. It proceeds by verifying that the polynomials satisfy the same divided-difference recurrences and initial conditions as the structure sheaves of the cells, using only the set-valued extension rules and the paving established internally in Section 2. No external geometric assumptions are invoked. We will insert a clarifying sentence at the start of the proof to emphasize this internal derivation, rendering the geometric interpretation of the truncations unconditional within the paper's framework. revision: partial

Circularity Check

0 steps flagged

No circularity; definition followed by independent duality proof

full rationale

The manuscript introduces grove polynomials via a combinatorial definition as a set-valued extension of forest polynomials, then establishes their K-theoretic duality to the quasisymmetric Schubert cells via explicit geometric and algebraic arguments that parallel the classical Grothendieck-Schubert case. The paving of the quasisymmetric flag variety is invoked as the ambient geometric setting rather than derived from the new polynomials themselves; the duality statement is shown to hold by direct comparison of classes and does not reduce to a self-referential equation or a fitted parameter renamed as a prediction. No load-bearing step collapses to a prior self-citation chain or an ansatz smuggled through citation; the derivation remains self-contained against the stated combinatorial and K-theoretic inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the new definition of grove polynomials and the domain assumption that the quasisymmetric flag variety is paved by the relevant cells; no free parameters or invented entities with independent evidence are apparent from the abstract.

axioms (1)
  • domain assumption The quasisymmetric flag variety is paved by quasisymmetric Schubert cells.
    This paving property is invoked to mirror the classical Schubert cell paving and enable the stated duality.
invented entities (1)
  • Grove polynomials no independent evidence
    purpose: Set-valued extension of forest polynomials serving as K-theoretic duals.
    Newly defined combinatorial object introduced to establish the duality.

pith-pipeline@v0.9.0 · 5608 in / 1267 out tokens · 61586 ms · 2026-05-22T03:43:58.462015+00:00 · methodology

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Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Aluffi, L

    P . Aluffi, L. C. Mihalcea, J. Schürmann, and C. Su. Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem.Ann. Sci. Éc. Norm. Supér. (4), 57(1):87–141, 2024. 15

    work page 2024

  2. [2]

    Anderson and W

    D. Anderson and W. Fulton.Equivariant cohomology in algebraic geometry, volume 210 ofCambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 2024. 3

    work page 2024

  3. [3]

    Bergeron, L

    N. Bergeron, L. Gagnon, P . Nadeau, H. Spink, and V . Tewari. The quasisymmetric flag variety: a toric complex on noncrossing partitions, 2025. 1, 14, 15

    work page 2025

  4. [4]

    Bergeron and F

    N. Bergeron and F. Sottile. Schubert polynomials, the Bruhat order, and the geometry of flag manifolds.Duke Math. J., 95(2):373–423, 1998. 15

    work page 1998

  5. [5]

    S. C. Billey, Y. Gao, and B. Pawlowski. Introduction to the Cohomology of the Flag Variety, 2025. 3

    work page 2025

  6. [6]

    A. Borel. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2), 57:115–207, 1953. 14

    work page 1953

  7. [7]

    M. Brion. Lectures on the geometry of flag varieties. InTopics in cohomological studies of algebraic varieties, Trends Math., pages 33–85. Birkhäuser, Basel, 2005. 3

    work page 2005

  8. [8]

    A. S. Buch. A Littlewood-Richardson rule for theK-theory of Grassmannians.Acta Math., 189(1):37–78, 2002. 16

    work page 2002

  9. [9]

    Demazure

    M. Demazure. Désingularisation des variétés de Schubert généralisées.Ann. Sci. École Norm. Sup. (4), 7:53–88, 1974. 14

    work page 1974

  10. [10]

    Fomin and A

    S. Fomin and A. N. Kirillov. Grothendieck polynomials and the Yang-Baxter equation. InFormal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, pages 183–189. DIMACS, Piscataway, NJ, sd. 1

    work page

  11. [11]

    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. 3

    work page 1997

  12. [12]

    Fulton and A

    W. Fulton and A. Lascoux. A Pieri formula in the Grothendieck ring of a flag bundle.Duke Math. J., 76(3):711–729,

    work page

  13. [13]

    I. M. Gessel. MultipartiteP-partitions and inner products of skew Schur functions. InCombinatorics and algebra (Boulder, Colo., 1983), volume 34 ofContemp. Math., pages 289–317. Amer. Math. Soc., Providence, RI, 1984. 2, 16

    work page 1983

  14. [14]

    Knutson and E

    A. Knutson and E. Miller. Gröbner geometry of Schubert polynomials.Ann. of Math. (2), 161(3):1245–1318, 2005. 1

    work page 2005

  15. [15]

    Lam and P

    T. Lam and P . Pylyavskyy. Combinatorial Hopf algebras andK-homology of Grassmannians.Int. Math. Res. Not. IMRN, (24):Art. ID rnm125, 48 pp, 2007. 2, 6, 17, 18

    work page 2007

  16. [16]

    A. Lascoux. Transition on Grothendieck polynomials. InPhysics and combinatorics, 2000 (Nagoya), pages 164–179. World Sci. Publ., River Edge, NJ, 2001. 2

    work page 2000

  17. [17]

    A. Lascoux. Schubert & Grothendieck: un bilan bidécennal.Sém. Lothar. Combin., 50:Art. B50i, 32, 2003/04. 13

    work page 2003

  18. [18]

    Lascoux and M.-P

    A. Lascoux and M.-P . Schützenberger. Polynômes de Schubert.C. R. Acad. Sci. Paris Sér. I Math., 294(13):447–450,

    work page

  19. [19]

    Lascoux and M.-P

    A. Lascoux and M.-P . Schützenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux.C. R. Acad. Sci. Paris Sér. I Math., 295(11):629–633, 1982. 1

    work page 1982

  20. [20]

    C. Lenart. Noncommutative Schubert calculus and Grothendieck polynomials.Adv. Math., 143(1):159–183, 1999. 13

    work page 1999

  21. [21]

    Lenart, S

    C. Lenart, S. Robinson, and F. Sottile. Grothendieck polynomials via permutation patterns and chains in the Bruhat order.Amer. J. Math., 128(4):805–848, 2006. 15 GROVE POLYNOMIALS ANDK-THEORETIC QUASISYMMETRY 19

    work page 2006

  22. [22]

    Monical, O

    C. Monical, O. Pechenik, and D. Searles. Polynomials from combinatorialK-theory.Canad. J. Math., 73(1):29–62,

    work page

  23. [23]

    Nadeau, H

    P . Nadeau, H. Spink, and V . Tewari. The geometry of quasisymmetric coinvariants, 2024. 9

    work page 2024

  24. [24]

    Nadeau, H

    P . Nadeau, H. Spink, and V . Tewari. Quasisymmetric divided differences, 2024. 1, 4, 13, 16, 17

    work page 2024

  25. [25]

    Nadeau and V

    P . Nadeau and V . Tewari. Forest polynomials and the class of the permutahedral variety.Adv. Math., 453:Paper No. 109834, 33, 2024. 1, 4, 6, 16

    work page 2024

  26. [26]

    Pechenik and D

    O. Pechenik and D. Searles. Decompositions of Grothendieck polynomials.Int. Math. Res. Not. IMRN, (10):3214– 3241, 2019. 2, 13

    work page 2019

  27. [27]

    Pechenik and D

    O. Pechenik and D. Searles. Asymmetric function theory. InSchubert calculus and its applications in combinatorics and representation theory, volume 332 ofSpringer Proc. Math. Stat., pages 73–112. Springer, Singapore, [2020] ©2020. 3

    work page 2020

  28. [28]

    Grothendieck to Lascoux

    V . Reiner and A. Yong. The "Grothendieck to Lascoux" conjecture, 2021. 13

    work page 2021

  29. [29]

    Shimozono and T

    M. Shimozono and T. Yu. Grothendieck-to-Lascoux expansions.Trans. Amer. Math. Soc., 376(7):5181–5220, 2023. 2, 13

    work page 2023

  30. [30]

    Weigandt

    A. Weigandt. Changing bases with pipe dream combinatorics, 2025. 13

    work page 2025

  31. [31]

    Woo and A

    A. Woo and A. Yong. Schubert geometry and combinatorics, 2023. 3 UNIVERSITÉLYON1, CENTRALELYON, INSA LYON, UNIVERSITÉJEANMONNET, CNRS, ICJ UMR5208, 69622 VILLEURBANNE, FRANCE Email address:nadeau@math.univ-lyon1.fr DEPARTMENT OFMATHEMATICS, UNIVERSITY OFTORONTO, TORONTO, ON M5S 2E4, CANADA Email address:hunter.spink@utoronto.ca DEPARTMENT OFMATHEMATICAL A...

    work page 2023