Pith. sign in

REVIEW 1 major objections 4 minor 31 references

Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper introduces weighted K-k-Schur functions — a family interpolating between K-k-Schur functions and closed k-Schur Katalan functions — and uses them to prove the K-k-Schur alternating conjecture for all partitions in a class that…

desk verdict A genuinely new interpolation family and a partial resolution of a real conjecture, but the proof of the key Proposition 3.6 has an unverified step that must be patched. read the letter →

arxiv 2507.23222 v1 pith:3VGM76GV submitted 2025-07-31 math.CO

Yaozhou Fan , Xing Gao This is my paper · ORCID
classification math.CO MSC 05E0505E1014N15
keywords KatalanfunctionweightedK-k-Schurclosedk-SchuralternatingconjecturepositivityK-theoreticSchubertcalculusk-boundedpartitionsMirrorLemma
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

This paper is trying to prove that a specific alternating sign pattern holds when closed k-Schur Katalan functions are expanded in the basis of K-k-Schur functions. The authors introduce a one-parameter interpolation between those two families, the weighted K-k-Schur functions, and show by induction on the weight that the expansion coefficients alternate in sign according to the size difference of the partitions. This resolves the K-k-Schur alternating conjecture for a large class of k-bounded partitions, including all strictly decreasing ones. The relevance is that such alternating positivity is the combinatorial backbone of K-theoretic Schubert calculus, where these symmetric functions represent geometric classes.

What carries the argument

The key object is the weighted K-k-Schur function $g^{(k)}_{\lambda,z}$, defined as the Katalan function $K(\Delta_k(\lambda); L(\Delta_k(\lambda))\setminus\{d_\lambda(x): x\in [z,b_\lambda]\}; \lambda)$. It is a Katalan function: an inhomogeneous symmetric function built from the dual stable Grothendieck determinant $g_\gamma$ by applying lowering operators $L_z$ and raising operators $R_{ij}$ encoded in a root ideal. For $z=1$ this is the K-k-Schur function $g^{(k)}_\lambda$; for $z=b_\lambda+1$ it is the closed k-Schur Katalan function $\tilde g^{(k)}_\lambda$. The argument runs on two levers: the recursion $g^{(k)}_{\lambda,z+1}=g^{(k)}_{\lambda,z}-L_{d_\lambda(z)}g^{(k)}_{\lambda,z}$, and the Mirror Lemma of Katalan-function theory, which makes many lowering-operator terms vanish or collapse. Proposition 3.6 converts these local cancellations into an expansion of weight $z+1$ functions in weight $z$ functions with alternating-positive coefficients, and induction on $z$ carries the positivity to the endpoint.

What would settle it

Compute the expansion $\tilde g^{(k)}_\lambda=\sum_\mu b_{\lambda\mu}g^{(k)}_\mu$ for all $\lambda\in\hat P^k_\ell$ with small $k$ and $\ell$ (say $k\le 7$, $\ell\le 8$) by applying the defining Katalan relations symbolically; the theorem is false if any $(-1)^{|\lambda|-|\mu|}b_{\lambda\mu}$ comes out negative or non-integral, and the paper's own Example 4.1 with $k=5$, $\lambda=(5,4,3,3,2,2)$ is one point where the signs all match.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.3: for every partition $\lambda\in \hat P^k_\ell$, the closed k-Schur Katalan function $\tilde g^{(k)}_\lambda$ expands as $\tilde g^{(k)}_\lambda = \sum_{\mu\in P} b_{\lambda\mu} g^{(k)}_\mu$, and the coefficients satisfy $(-1)^{|\lambda|-|\mu|} b_{\lambda\mu}\in \mathbb{Z}_{\ge 0}$. Here $\hat P^k_\ell$ is the class of k-bounded partitions with $\lambda_{x-1}>\lambda_x$ whenever $k-\lambda_x+x<\ell$, which includes all strictly decreasing k-bounded partitions. The proof goes through the stronger Theorem 1.2: under the strictness hypothesis $\lambda_1>\cdots>\lambda_z$ (and $\lambda_z>\lambda_{z+1}$ when $z\neq b_\lambda$), the weighted function $g^{(k)}_{\lambda,z+1}$ has an alternating-positive expansion in the weight-$z$ functions $g^{(k)}_{\mu,z}$. Since $z=1$ recovers $g^{(k)}_\lambda$ and $z=b_\lambda+1$ recovers $\tilde g^{(k)}_\lambda$, the theorem interpolates from one side of the conjecture to the other.

Load-bearing premise

The proof depends on the partition's first entries being strictly decreasing at each step of the recursion, and when that strictness fails the Mirror Lemma cannot be applied and the expansion is not proved.

Editorial extensions

If this is right

  • Conjecture 1.1(e) is now a theorem for every partition in $\hat P^k_\ell$, so the closed k-Schur Katalan function of any such partition is alternating-positive in the K-k-Schur basis.
  • Every strictly decreasing k-bounded partition lies in $\hat P^k_\ell$, so the conjecture holds for all strictly decreasing k-bounded partitions.
  • The weighted family $g^{(k)}_{\lambda,z}$ gives a chain of alternating-positive expansions interpolating between $g^{(k)}_\lambda$ and $\tilde g^{(k)}_\lambda$; passing from weight $z$ to weight $z+1$ never destroys the sign pattern.
  • Theorem 1.2 supplies a recursive algorithm for computing the coefficients $b_{\lambda\mu}$: expand $g^{(k)}_{\lambda,z+1}$ via Proposition 3.6, then expand each weight-$z$ term by induction down to $z=1$.
  • The sign twist $(-1)^{|\lambda|-|\mu|}b_{\lambda\mu}$ is a nonnegative integer, so each signed coefficient is a whole number with a definite sign.

Reading between the lines

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

  • Editorial inference: The weight parameter $z$ can be read as a filtration interpolating between two geometric bases; a $q$-analogue weighting each step by $q^z$ would produce a $q$-K-k-Schur function whose specialization at $q=0$ and $q=1$ gives the two extremes. The paper does not define such an object.
  • Editorial inference: If alternating positivity holds outside $\hat P^k_\ell$ as well, the obstruction is not the positivity itself but the Mirror Lemma's strictness hypotheses; a search for the smallest partition with a repeated part in the initial segment would test whether the theorem's class is sharp.
  • Editorial inference: The same induction, applied to other Katalan-function identities, may transfer these methods to the related alternating dual Pieri and k-branching conjectures, since those conjectures are logically linked in the same framework.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper introduces a new family of Katalan functions, the weighted K-k-Schur functions g^{(k)}_{λ,z}, which interpolate between K-k-Schur functions (z=1) and closed k-Schur Katalan functions (z=b_λ+1). The central result is Proposition 3.6, a recursive expansion of g^{(k)}_{λ,z+1} in terms of g^{(k)}_{μ,z} with alternating positive coefficients, and its consequences: Theorems 1.2 and 1.3, which prove the K-k-Schur alternating conjecture (Conjecture 1.1(e) of Blasiak–Morse–Seelinger) for the class \hat P^k_ℓ of partitions satisfying strict decrease up to the bounce bottom b_λ, including all strictly decreasing k-bounded partitions. The proofs use the Katalan formula, the Mirror Lemma, and a delicate induction on the bounce path of λ.

Significance. If the proof is completed, the result is a substantial advance: it resolves a named conjecture from the 2022 work of Blasiak, Morse, and Seelinger for a natural and fairly broad class of partitions, and it introduces an interpolating family that is likely to be useful for the remaining conjectures in that program. The paper is careful and mostly self-contained: it gives explicit definitions, worked examples, and detailed verifications of the Mirror Lemma hypotheses in the key propositions. It relies on the externally established Katalan formula and Mirror Lemma, and the new weighted functions are defined rather than fitted to the conclusion, so there is no evident circularity. The main concern is a genuine gap in the central induction, described below; because the gap is local and the surrounding argument is plausible, the manuscript merits revision rather than rejection.

major comments (1)
  1. [§3.2, Proposition 3.6, display (24)] The induction hypothesis (21) is applied to the auxiliary partition γ without verifying that the lowering operator L_{d^{c'-(c-1)}_γ(z)} is defined and lies on γ's own bounce path. This is not automatic: for k=5, ℓ=6, λ=(5,2,1,1,1,1), z=1, one has c'=2 and, at c=1, γ=λ−ε_{d^1_λ(1)}=(5,1,1,1,1,1) has b_γ=1, so d^2_γ(1) is undefined. Proposition 3.4 has a convention setting such a lowering to 0, but Proposition 3.6 does not state that this convention applies in display (24); as written, (24) asserts an expansion for an operator that is not defined. This occurs at the final inductive step (c=1, a=c'-1) and is exactly the step on which the proofs of Theorems 1.2 and 1.3 depend. The expansion in the example appears to be correct, so the gap is likely repairable, but the proof must be amended: either the undefined case must be treated separately using the convention from Proposition 3.4, or one must prove that when the operator is defined, its exponent lies in the range of γ's own bounce path before applying (21).
minor comments (4)
  1. [§3.2, Proposition 3.6 proof] The claim that d^{c'-c}_λ(z) ≠ z, z+1 is not generally true; for instance, with k=5, λ=(5,4,3,3,3,3), z=1, one has d^1_λ(1)=2=z+1. The needed conclusion γ_z>γ_{z+1} still follows from λ_z>λ_{z+1} in such cases, but the stated reason is inaccurate and should be corrected.
  2. [§3.1, Proposition 3.4] The reference "Remark 2.8 (b)" should be "Remark 2.8 (2)".
  3. [§3.2, display (24)] Display (24) omits the membership "∈ Z_{≥0}" after the alternating coefficient; the text uses this membership immediately afterward and should include it.
  4. [Example 3.7] The example writes the partition (7,6,5,4,4,4,4,3,3,3,2,2,0) with a trailing zero part; standard partition notation would omit the zero, which could confuse readers.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.3 is derived from external Katalan-function machinery, and the authors' own prior work is cited only for context, not as the load-bearing input.

full rationale

The central claim, Theorem 1.3, is not assumed as an input. The weighted K-k-Schur functions in Definition 2.12 interpolate between g^{(k)}_lambda and tilde-g^{(k)}_lambda by Proposition 2.13, but that interpolation is definitional and does not itself encode the alternating expansion. The recursive decomposition in Proposition 3.6 is obtained from the Katalan-function expansion rules (Lemma 2.5) and the Mirror Lemma of Blasiak-Morse-Seelinger [4], which are external to this paper. Theorem 1.2 then iterates that recursion with explicit coefficient-sign bookkeeping, and Theorem 1.3 is a direct corollary for the class in (1). The paper's own prior work [8,9] is cited for reporting earlier proofs of other parts of Conjecture 1.1 and for elementary root-ideal facts that are also attributed to [4]; none of these self-citations is load-bearing for Theorems 1.2-1.3. I therefore find no circular reduction. I do flag, as a proof-completeness concern rather than a circularity, that in the proof of Proposition 3.6 display (24) applies the induction hypothesis (21) to gamma = lambda - epsilon_{d^{c'-c}_lambda(z)} without explicitly verifying that d^{c'-(c-1)}_gamma(z) lies on gamma's own bounce path; if that operator is undefined, the stated expansion is not covered by (21). This is a potential gap in the derivation, but it does not make the theorem equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the Katalan framework of Blasiak-Morse-Seelinger: the Katalan formula and Mirror Lemma are external results used as black boxes. The only new invented object is the weighted K-k-Schur function, which is a definition, not a postulate. There are no fitted constants; k and z are discrete parameters of the construction.

assumptions (4)
  • domain assumption Mirror Lemma ([4, Lemma 4.6])
    Used in Propositions 3.2 and 3.4 to force alternating terms to vanish or to reduce a lowering operator; the proof of Theorem 1.2 depends on it.
  • domain assumption Katalan formula for K-k-Schur functions ([4, Theorem 2.6], Lemma 2.9 in this paper)
    Gives the expression g^{(k)}_{\lambda} = K(\Delta_k(\lambda); \Delta_{k+1}(\lambda); \lambda) used as the starting point.
  • standard math Root ideal, bounce path, and bottom definitions from [4]
    Reused without proof as background combinatorial machinery.
  • standard math Standard ZFC and elementary linear algebra over symmetric functions
    The proof is a finite induction; no special foundational assumptions are made.
invented entities (1)
  • Weighted K-k-Schur function g^{(k)}_{\lambda,z}
    purpose: Interpolates between K-k-Schur functions (z=1) and closed k-Schur Katalan functions (z=b_\lambda+1), enabling an inductive proof of alternating positivity.
    This is a new mathematical object defined explicitly in Definition 2.12; it is not an empirically postulated entity but a construction. Its properties are proven within the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture." pith.science (2026). https://pith.science/paper/3VGM76GV

@misc{pith2026250723222,
  author       = {Pith},
  title        = {Pith review of: Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VGM76GV}},
  note         = {Machine review of arXiv:2507.23222}
}
abstract

We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed $k$-bounded partitions. As a central application, we resolve the $K$-$k$-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of $k$-bounded partitions, including all strictly decreasing $k$-bounded partitions. Our results shed new light on the combinatorial structure of $K$-theoretic symmetric functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Blasiak, J

    J. Blasiak, J. Morse and A. Pun, Demazure crystals and the Schur positivity of Catalan functions,Invent. math. 236 (2024), 483-547. 2

    work page 2024

  2. [2]

    Blasiak, J

    J. Blasiak, J. Morse, A. Pun and D. Summers, Catalan functions and k-schur positivity, J. Amer. Math. Soc.32 (4) (2019), 921-963. 2, 3, 4, 6

    work page 2019

  3. [3]

    Blasiak, J

    J. Blasiak, J. Morse, A. Pun and D. Summers, k-Schur expansions of Catalan functions,Adv. Math.371 (2020), 107209. 2, 4

    work page 2020

  4. [4]

    Blasiak, J

    J. Blasiak, J. Morse and G. H. Seelinger, K-theoretic Catalan functions, Adv. Math. 404 (2022), 108421. 1, 2, 3, 4, 5, 6, 7, 8

    work page 2022

  5. [5]

    Bott, The space of loops on a Lie group, Michigan Math

    R. Bott, The space of loops on a Lie group, Michigan Math. J. 5(1) (1958), 35-61. 2

    work page 1958

  6. [6]

    Broer, Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, Lie theory and geometry, Progr

    B. Broer, Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, Lie theory and geometry, Progr. Math., Vol. 123, Birkh¨auser Boston, Inc., Boston, MA, 1994, 1-19. 2

    work page 1994

  7. [7]

    L. C. Chen, Skew-Linked Partitions and a Representation-Theoretic Model for k-Schur Functions, ProQuest LLC, Ann Arbor, MI, 2010, Thesis (Ph.D.)-University of California, Berkeley. 2

    work page 2010

  8. [8]

    Y . Z. Fang and X. Gao, Alternating dual Pieri rule conjecture and k-branching conjecture of closed k-Schur Katalan functions, arXiv:2501.04200. 3, 4, 8

    work page Pith review arXiv

Show all 31 references
  1. [9]

    Y . Z. Fang, X. Gao and L. Guo, Lowering operators on K-k-Schur functions and a combinatorial proof of a conjecture on closed k-Schur Katalan functions, arXiv:2502.05618. 3, 4

    arXiv

  2. [10]

    Fomin, S

    S. Fomin, S. Gelfand and A. Postnikov, Quantum Schubert polynomials, J. Am. Math. Soc. 10(3) (1997) 565-

    1997

  3. [11]

    Givental and Y

    A. Givental and Y . P. Lee, Quantum K-theory on flag manifolds, finite-di fference Toda lattices and quantum groups, Invent. Math. 151(1) (2003), 193-219. 3

    2003

  4. [12]

    Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Com- binatorics of Macdonald Polynomials, American Mathematical Society, 2007

    J. Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Com- binatorics of Macdonald Polynomials, American Mathematical Society, 2007. 2

    2007

  5. [13]

    Hilbert, Mathematical problems, Bull

    D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437-479. 2

    1902

  6. [14]

    Ikeda, S

    T. Ikeda, S. Iwao and T. Maeno, Peterson isomorphism in K-theory and relativistic Toda lattice,Int. Math. Res. Not. 2020(19) (2020), 6421-6462. 2, 3

    2020

  7. [15]

    Ikeda, S

    T. Ikeda, S. Iwao and S. Naito, Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian, Trans. Amer. Math. Soc. Ser. B11(20) (2024), 667-702. 3

    2024

  8. [16]

    Lam, Schubert polynomials for the a ffine Grassmannian, J

    T. Lam, Schubert polynomials for the a ffine Grassmannian, J. Amer. Math. Soc.21(1) (2008), 259-281. 2

    2008

  9. [17]

    T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono and M. Zabrocki, k-Schur functions and a ffine Schubert calculus, Fields Institute Monographs, vol. 33, Springer/Fields Institute for Research in Mathematical Sciences, New York/Toronto, 2014. 2

    2014

  10. [18]

    T. Lam, C. Li, L. C. Mihalcea and M. Shimozono, A conjectural Peterson isomorphism inK-theory, J. Algebra 513 (2018), 326-343. 2

    2018

  11. [19]

    T. Lam, A. Schilling and M. Shimozono, K-theory schubert calculus of the a ffine grassmannian, Compos. Math. 146(4) (2010), 811-852. 2, 8

    2010

  12. [20]

    Lam and M

    T. Lam and M. Shimozono, From quantum Schubert polynomials to k-Schur functions via the Toda lattice, Math. Res. Lett. 19(1) (2012), 81-93. 2, 3

    2012

  13. [21]

    Lapointe, A

    L. Lapointe, A. Lascoux and J. Morse, Tableau atoms and a new Macdonald positivity conjecture,Duke Math. J. 116(1) (2003), 103-146. 2

    2003

  14. [22]

    Lapointe and J

    L. Lapointe and J. Morse, A k-tableau characterization of k-Schur functions, Adv. Math. 213(1) (2007), 183-

    2007

  15. [23]

    A. Lascoux, Ordering the a ffine symmetric group, Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in G¨ oßweinstein, Germany, September 12–19, 1999 90(7) (2001), 219-231. 2, 5

    2001

  16. [24]

    Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann

    C. Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb. 4 (2000), 67-82. 2, 5

    2000

  17. [25]

    Lenart and T

    C. Lenart and T. Maeno, Quantum Grothendieck polynomials, arXiv:math /0608232. 3

  18. [26]

    Morse, Combinatorics of the k-theory of affine grassmannians, Adv

    J. Morse, Combinatorics of the k-theory of affine grassmannians, Adv. Math. 229 (2012), 2950-2984. 8

    2012

  19. [27]

    D. I. Panyushev, Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles, Selecta Math. (N.S.) 16(2) (2010), 315-342. 2

    2010

  20. [28]

    Schubert, Kalk ¨ul der abz ¨ahlenden Geometrie (German) (Calculus of enumerative geometry), Springer- Verlag, Berlin-New York, 1979

    H. Schubert, Kalk ¨ul der abz ¨ahlenden Geometrie (German) (Calculus of enumerative geometry), Springer- Verlag, Berlin-New York, 1979. Reprint of the 1879 original, With an introduction by Steven L. Kleiman. 2

    1979

  21. [29]

    Shimozono and J

    M. Shimozono and J. Weyman, Graded characters of modules supported in the closure of a nilpotent conjugacy class, European J. Combin. 21(2) (2000), 257-288. 2

    2000

  22. [30]

    Takigiku, A Pieri formula and a factorization formula for sums of K-theoretic k-Schur functions, Algebr

    M. Takigiku, A Pieri formula and a factorization formula for sums of K-theoretic k-Schur functions, Algebr. Comb. 2(4) (2019), 447-480. 3 School of Mathematics andStatistics, Lanzhou University Lanzhou, 730000, China Email address: fangyzh21@lzu.edu.cn School of Mathematics an...

    2019

  23. [204]

    3 K-k-SCHUR ALTERNATING CONJECTURE 23

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.