Grothendieck positivity for normal square root crystals

Eric Marberg; Kam Hung Tong; Tianyi Yu

REVIEW 3 major objections 4 minor 1 cited by

Grothendieck positivity for normal square root crystals

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Every finite normal square root crystal has a character that is a sum of symmetric Grothendieck polynomials.

desk verdict Strong and interesting framework, but the proof of the central positivity theorem relies on a Hecke-insertion lemma that is false as stated. read the letter →

arxiv 2501.16640 v2 pith:BVAQ45P2 submitted 2025-01-28 math.CO math.RT

Eric Marberg , Kam Hung Tong , Tianyi Yu This is my paper · ORCID

classification math.COmath.RT MSC 05E0505E1017B37

keywords normalsquarerootcrystalsGrothendieckpolynomialsHeckeinsertionStembridgerectificationset-valuedtableauxcrystalcharacterK-theoreticcombinatorics

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

Normal square root crystals are a K-theoretic analogue of Stembridge crystals: their characters are meant to expand positively in the basis of symmetric Grothendieck polynomials instead of Schur polynomials. The paper proves that this is always true: for every finite normal square root crystal, the character equals the sum of symmetric Grothendieck polynomials attached to its highest weight elements. This settles a conjecture from the paper that introduced these crystals and gives a uniform mechanism for proving Grothendieck positivity of symmetric functions, just as normal crystals give Schur positivity. The proof works by showing that the rectification operator on such crystals is realized by Hecke insertion.

What carries the argument

The central object is the rectification operator rect, defined as the composition of iterated raising operators applied in a specific order, together with its realization by Hecke insertion. For each set-valued word S, Theorem 3.11 proves that the Hecke-insertion recording tableau P_Hecke(S) equals tab(rect(S)), where tab rewrites a set-valued word as a tableau. The load-bearing technical bridge is Lemma 3.6, which asserts that Hecke insertion of a decreasing set of numbers into a multirow increasing tableau reduces to inserting them into the last row and then reinserting the bumped entries into the remaining rows; this reduction is what lets the proof compare the crystal-raising process with the insertion algorithm dimension by dimension.

What would settle it

Test Lemma 3.6 by hand on a three-row increasing tableau with a decreasing set of inserted numbers; for example, take P = 1 2 5 / 3 4 6 / 7 and B = {6,4,2}, and compare direct Hecke insertion with the lemma's reduction to last-row insertion.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: if B is a finite normal sqrt(gln)-crystal — normal meaning each connected component is isomorphic to a full subcrystal of a tensor power of the standard sqrt(gln)-crystal on nonempty subsets of {1,...,n} — then ch(B) = sum_{b in HW(B)} G_{wt(b)}(x_1,...,x_n). In words, the character of any finite normal square root crystal is a sum of symmetric Grothendieck polynomials, one per highest weight element. This is the exact K-theoretic analogue of the classical fact that characters of normal gln-crystals are Schur-positive. The proof establishes a bijection between elements of a normal square root crystal and pairs (P,Q) consisting of an increasing tableau P indexed by a highest weight element and a semistandard set-valued tableau Q of the same shape, mediated by Hecke insertion.

Load-bearing premise

The load-bearing premise is Lemma 3.6, which asserts that Hecke insertion of a decreasing set into a multirow increasing tableau reduces to insertion into the last row followed by reinsertion into the remaining rows; if this reduction fails, the paper's proof of the main theorem collapses.

Editorial extensions

If this is right

Theorem 1.1 yields a new proof of the Littlewood–Richardson rule for multiplying symmetric Grothendieck polynomials, by applying the theorem to the normal square root crystal on set-valued tableaux of skew shape.
The Grothendieck polynomial of a permutation is Grothendieck-positive, with coefficients counted by increasing tableaux whose reverse row reading word is a Hecke word for the permutation (Corollary 1.11).
The generating function for set-valued decomposition tableaux of a strict partition is Grothendieck-positive (Corollary 1.13).
Every homogeneous piece of the character of a normal square root crystal is Schur positive (Corollary 3.15), so square root crystals refine ordinary Schur positivity.
The rectification operator sends every element of a normal square root crystal to a highest weight element (Theorem 2.21), the fact that makes the highest-weight summation formula possible.

Reading between the lines

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

The rectification-to-Hecke-insertion correspondence suggests a broader template: any crystal family whose raising operators can be simulated by a tableau insertion algorithm should have characters that expand positively in the associated Grothendieck-type basis; testing this on other K-theoretic crystals would show how general the phenomenon is.
The paper leaves open the search for local Stembridge-style axioms that characterize normal square root crystals; the main theorem makes such axioms more valuable, since they would give an effective way to recognize when a generating function is Grothendieck-positive.
The authors verified a Lascoux-positivity conjecture for square root Demazure crystals by computer for all m,n <= 5 except (5,5); a natural next test is to check that remaining case and to see whether the same Hecke-insertion technology can prove it uniformly.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main thing to know: this is a substantial paper with a real result in reach, but there is a load-bearing error in the proof. Theorem 1.1—that finite normal sqrt(gl_n)-crystals have Grothendieck-positive characters—is exactly the kind of structural bridge people have wanted since [MT23], and the applications (Buch's rule, G-positivity of Gw, Corollary 1.13) make it worth reading. The writing is clear and the overall architecture is sensible: rectify to highest weight elements, then identify those with Hecke insertion tableaux.

The soft spot is not a stylistic issue. Lemma 3.5(c) is false. Let T = [[1,3],[2]] and x = 1. T is a two-row increasing tableau, not a rectangle, and T_{2,1} = 2 >= T_{1,1}+1 = 2. The lemma predicts (1 H-> T) = T. Running Definition 3.1: inserting 1 into column 1 leaves the column unchanged and bumps 2; inserting 2 into column 2 replaces 3 and bumps 3; inserting 3 into the empty column 3 appends it. The result is [[1,2,3],[2]] != T. This false statement is not decorative: Lemma 3.12, Case 3, uses Lemma 3.5(c) to conclude that Hecke inserting the top-left entry fixes tab(E1(S1)) in the combined-form subcase. Lemma 3.12 feeds Lemma 3.13, Theorem 3.11, and ultimately Theorem 1.1. So the written proof has a concrete gap at a load-bearing step. The theorem may still be true—the applications are consistent and external results support the framework—but the derivation as written is not valid there.

Other soft spots are minor by comparison: Lemma 3.5 is stated as a 'basic exercise' without proof, which turned out to be a bad bet; Lemma 3.6 is a long case analysis and not machine-checked; Example 1.14's bijection is shown by example rather than formally constructed. None of these matter as much as the false lemma.

Who this is for: anyone working in K-theoretic Schubert calculus or crystal theory. The framework is novel and the intended applications are the right ones. But I would not trust the main theorem until Lemma 3.5/3.12 is repaired. I would send the paper to a serious referee—there is enough substance that the error may be fixable and the paper deserves a full report—but my own verdict on the current version is skeptical.

Referee Report

3 major / 4 minor

Summary. The paper studies finite normal square root crystals for gl_n and proves that the character of any such crystal is a sum of symmetric Grothendieck polynomials indexed by its highest weight elements (Theorem 1.1). The proof develops a rectification operator for square root crystals, connects it to the Hecke insertion algorithm of Buch, Kresch, Shimozono, Tamvakis, and Yong, and derives a bijection between full subcrystals and pairs of tableaux (P,Q). The introduction presents several applications: a new proof of Buch's combinatorial rule for skew symmetric Grothendieck polynomials, a K-theoretic Littlewood-Richardson rule, G-positivity for permutation Grothendieck polynomials, and a new positivity statement for set-valued decomposition tableaux.

Significance. If Theorem 1.1 is correct, it is a substantial contribution: it resolves a conjecture of Marberg and Tong and provides a uniform combinatorial mechanism for proving Grothendieck positivity, analogous to the role of Stembridge crystals for Schur positivity. The connection between square root crystals and Hecke insertion is elegant, and the applications in Section 1 are natural and illustrate the power of the framework. The paper is generally well structured and the reliance on prior work is transparent. However, the written proof has a load-bearing gap in a technical lemma, so the main theorem is not established by the current text.

major comments (3)

[§3.1, Lemma 3.5(c)] Lemma 3.5(c) is false under Definition 3.1. Take T = [[1,3],[2]] (English notation: first row 1,3; second row 2) and x = 1. Then T is an increasing tableau, it is not a rectangle, and for the only second-row box we have T_{2,1} = 2 >= T_{1,1}+1 = 2, so the hypotheses of (c) hold. Following Definition 3.1, inserting 1 into the first column finds y = 2 and bumps it, since replacing 2 by 1 would violate strict increase. Inserting the bumped value 2 into the second column replaces 3, which is valid because the resulting tableau is still increasing. The bumped 3 is then appended to a new third column. The final tableau is [[1,2,3],[2]], not T. Thus the lemma is false as stated.
[§3.2, Lemma 3.12, Case 3] The false clause (c) is load-bearing. In the combined-form subcase of Case 3, the proof reduces the desired equality to the assertion that Hecke-inserting m+1-j into tab(E1(S1)) leaves the tableau unchanged, and it states that this follows from Lemmas 2.20(b) and 3.5(c). The counterexample above satisfies exactly the same hypotheses used there, so this step is not justified. Since Lemma 3.12 is used to prove Theorem 3.11, and Theorem 3.11 is the bridge between rectification and Hecke insertion used in Theorem 3.14 and then Theorem 1.1, the main proof has a concrete gap. The main theorem may still be true, but the written derivation does not establish it.
[§3.1, Lemma 3.5] Lemma 3.5 is stated without proof and described only as following from the definitions as a basic exercise. It is not a basic exercise, and in its current form it is incorrect. Because the lemma is invoked in the proof of Lemma 3.12, it needs either a correct proof or a repaired statement that still supports the combined-form argument.

minor comments (4)

[Example 1.14 / Figure 3] Example 1.14 defines k = max{1, m-n+1}, while the caption of Figure 3 gives k = max{1, m-n-1}; these should be reconciled.
[§3.1, Lemma 3.6] The proof of Lemma 3.6 is a long case analysis and is hard to check; a precise statement of the induction invariant, or at least a short overview of the induction, would substantially improve readability.
[Definition 3.1] The wording 'bumps y' in the case where y is not actually replaced is potentially confusing; a sentence clarifying that the bumped value is passed to the next column regardless of whether the replacement was valid would help.
[Proposition 3.10] The bijectivity of the map S -> (A(S), I(S)) is labeled a straightforward exercise; a brief proof sketch would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main G-positivity theorem is derived from the external Hecke insertion bijection and prior independent crystal constructions; the paper's self-citations are not load-bearing in a circular sense.

full rationale

The paper's central claim, Theorem 1.1, is not assumed as an input. The derivation chain is: define normal square root crystals via tensor powers of the standard crystal (Definition 2.7); identify normal components with full subcrystals of set-valued words (Section 2.2, citing Yu23 and MT23); prove rectification sends every element to a highest weight element (Theorem 2.21, proved self-contained via Lemmas 2.23-2.27); prove P_Hecke(S)=tab(rect(S)) (Theorem 3.11) using the Hecke insertion algorithm from BKS+08 and the technical Lemmas 3.5, 3.6, 3.12, and 3.13; then conclude that the character is a sum over highest weights of symmetric Grothendieck polynomials. The Hecke insertion bijection is external to the paper, and the crystal structures from Yu23 and MT23 are proved in those separate papers and do not encode the target theorem. No fitted parameters or 'prediction from data' occur anywhere in the argument. Self-citations to the authors' prior work exist, but the load-bearing content (normality of set-valued words, highest-weight characterizations) is independently established in those cited papers, not derived from Theorem 1.1. The main theorem is also checked against external benchmarks: it recovers Buch's combinatorial rule (Corollary 1.4), Buch's Littlewood-Richardson rule (Corollary 1.6), and the BKS+08 G-positivity result (Corollary 1.11). Therefore no circular step can be exhibited. Separately, for correctness rather than circularity: Lemma 3.5 is stated with 'We omit its proof, which follows as a basic exercise from the definitions,' and the reviewer's counterexample suggests that lemma may be false; this is a proof-gap concern, not a circularity concern, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are introduced: all weights and shapes are structural. The central claim rests on external tools: the Hecke insertion bijection [BKS+08], the set-valued word model of normal square root crystals [Yu23, MT23], and standard facts about symmetric Grothendieck polynomials. These are domain assumptions drawn from the literature, not ad hoc assumptions invented for this paper.

assumptions (5)

domain assumption Hecke insertion is a bijection from compatible sequences (A,I) to pairs (P,Q) with P an increasing tableau and Q a semistandard set-valued tableau of the same shape.
Inherited from [BKS+08]; used in Theorems 3.11 and 3.14.
domain assumption SetWord_{m,n} admits a normal sqrt(gln)-crystal structure with crystal operators computed by the i-word pairing rule of [Yu23].
Used throughout Section 2 as the standard model for normal square root crystals.
domain assumption Every connected normal sqrt(gln)-crystal is isomorphic to a full subcrystal of SetWord_{m,n} for some m.
This is the definition of normality for square root crystals from [MT23], used in the proof of Theorem 1.1.
domain assumption The symmetric Grothendieck polynomial G_lambda is the weight generating function of semistandard set-valued tableaux of shape lambda, and each homogeneous component of G_lambda is Schur positive.
Standard facts from [Buc02, Len00, MPS21]; used in Definition 1.2, Theorem 1.1, and Corollary 3.15.
domain assumption Highest weight elements of SetWord_{m,n} are exactly those S for which tab(S) is an increasing tableau.
Proved as Proposition 2.17 using [MT23, Lem. 4.11]; underlies the description of highest weight elements throughout Section 3.

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Pith. "Pith review of Grothendieck positivity for normal square root crystals." pith.science (2026). https://pith.science/paper/BVAQ45P2

@misc{pith2026250116640,
  author       = {Pith},
  title        = {Pith review of: Grothendieck positivity for normal square root crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BVAQ45P2}},
  note         = {Machine review of arXiv:2501.16640}
}

abstract

Normal crystals (also known as Stembridge crystals) are commonly used to establish the Schur positivity of symmetric functions, as their characters are sums of Schur polynomials. In this paper, we develop a combinatorial framework for a novel family of objects called normal square root crystals, which are closely related to symmetric Grothendieck functions, the $K$-theoretic analogue of Schur functions. Among other applications, this tool leads to a new proof of Buch's combinatorial rule for the multiplication of symmetric Grothendieck functions. The definition of a normal square root crystal, originally formulated by the first two authors, largely mirrors that of normal crystals. Our main result is to show that the character of such a crystal is always a sum of symmetric Grothendieck polynomials. The proof relies on an unexpected connection between the raising operators for our crystals and the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis, and Yong.

Figures

Figures reproduced from arXiv: 2501.16640 by the authors.

**Figure 1.** The standard gln-crystal, for which the weight map is wtp i q “ ei P Z n . varieties in the complex Grassmannian [Buc02, Thm. 8.1], while Schur polynomials are representatives for the cohomology classes. Schur polynomials and symmetric Grothendieck polynomials have many other interpretations in representation theory and geometry. Because of these interpretations, it is often meaningful to prove that generating func… view at source ↗

**Figure 2.** The standard a gl3 -crystal, for which the weight map is wtpSq “ ř iPS ei P Z 3 . An element of gln - or a gln -crystal is highest weight if it is a source vertex in the associated crystal graph. Let HWpBq denote the set of highest weight elements in a crystal B. When B is normal, the weight of every element of HWpbq is guaranteed to be a partition with at most n parts; see [BS17, §2.4] for the gln-case and [MT23, L… view at source ↗

**Figure 3.** The polynomial GPpmq px1, . . . , xnq is the weight-generating function for all fillings T of a one-row Young diagram with m boxes by nonempty subsets of t1 1 ă 1 ă 2 1 ă 2 ă ¨ ¨ ¨ ă n 1 ă nu such that maxpT1j q ď minpT1,j`1q for all j P rm ´ 1s, with strict inequality if maxpT1j q is primed [IN13, §9.1]. One can inductively construct a weight-preserving bijection between such tableaux and the union Ům i“k SetTabpi,… view at source ↗

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

Hybrid Grothendieck polynomials
math.CO 2025-05 conditional novelty 7.0 of 10

Hybrid Grothendieck polynomials unify stable and dual stable Grothendieck polynomials, with proofs of symmetry, Schur expansion, saturated Newton polytopes, and an omega image formula.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

Combinatorics of C oxeter groups

Anders Bj\"orner and Francesco Brenti. Combinatorics of C oxeter groups . Springer, New York, 2005. Graduate Texts in Mathematics, No. 231

work page 2005

[2] [1]

R C -graphs and S chubert polynomials

Nantel Bergeron and Sara Billey. R C -graphs and S chubert polynomials. Experimental Mathematics , 2:257--269, 1993

work page 1993

[3] [3]

Stable G rothendieck polynomials and K -theoretic factor sequences

Anders Skovsted Buch, Andrew Kresch, Mark Shimozono, Harry Tamvakis, and Alexander Yong. Stable G rothendieck polynomials and K -theoretic factor sequences. Mathematische annalen , 340:359--382, 2008

work page 2008

[4] [4]

Crystal bases: representations of combinatorics

Daniel Bump and Anne Schilling. Crystal bases: representations of combinatorics . World Scientific, Singapore, 2017

work page 2017

[5] [5]

A L ittlewood- R ichardson rule for the K -theory of G rassmannians

Anders Skovste Buch. A L ittlewood- R ichardson rule for the K -theory of G rassmannians. Acta Math. , 189:37--78, 2002

work page 2002

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Balanced tableaux

Paul Edelman and Curtis Greene. Balanced tableaux. Adv. Math. , 63:42--99, 1987

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Kirillov

Sergey Fomin and Anatol N. Kirillov. Grothendieck polynomials and the Y ang- B axter equation. In Proc. formal power series and alg. comb , pages 183--190, 1994

work page 1994

[8] [8]

Characterization of queer supercrystals

Maria Gillespie, Graham Hawkes, Wencin Poh, and Anne Schilling. Characterization of queer supercrystals. J. Combin. Theory Ser. A , 173(105235), 2020

work page 2020