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Hybrid Grothendieck polynomials

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Pith's one-line read The paper introduces a hybrid Grothendieck polynomial over set-valued reverse plane partitions, proves its Schur expansion with reverse-lattice-word coefficients, shows straight shapes have saturated Newton polytopes, and gives a…

desk verdict Solid unification of stable and dual stable Grothendieck polynomials via a new crystal on SVRPPs; main results are likely correct, but a few load-bearing local checks are asserted rather than proved. read the letter →

arxiv 2505.19072 v1 pith:6KSVSC7K submitted 2025-05-25 math.CO math.AGmath.RT

Peter L. Guo , Mingyang Kang , Jiaji Liu This is my paper · ORCID
classification math.COmath.AGmath.RT MSC 05E0505E10
keywords hybridGrothendieckpolynomialsset-valuedreverseplanepartitionscrystalbasesSchurexpansionsaturatedNewtonpolytopesomegainvolutionFomin–Greeneoperatorslatticewords
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 two-parameter deformation of skew Schur functions, called the hybrid Grothendieck polynomial $G_{\lambda/\mu}(\mathbf{x};\mathbf{t};\mathbf{w})$, as the weight generating function over set-valued reverse plane partitions of skew shape $\lambda/\mu$, with weights tracking irredundant content, column equalities, and excess. Setting all $t_i=0$ recovers the refined stable Grothendieck polynomial, and setting all $w_i=0$ recovers the refined dual stable Grothendieck polynomial, so the new object is a common parent of the two K-theoretic families. The paper proves that $G_{\lambda/\mu}$ is symmetric in the $\mathbf{x}$ variables and expands into Schur functions with coefficients counted by set-valued reverse plane partitions whose reading word is a reverse lattice word. For straight shapes it shows the resulting polynomial has a saturated Newton polytope, and it gives a combinatorial formula for the image under the omega involution in the one-parameter specialization.

What carries the argument

The central objects are set-valued reverse plane partitions, fillings of a skew shape with nonempty finite sets of positive integers, weakly increasing along rows and columns, together with the three statistics ircont, ceq, and ex. The engine of the paper is an involution on two-letter set-valued reverse plane partitions obtained by flipping pure columns and then resolving descents by local rewrites on benign two-letter tables; this involution makes the crystal operators $e_i,f_i$ well-defined and independent of the order of resolution. The crystals are shown to be seminormal $\mathfrak{gl}_n$-crystals whose highest-weight elements are exactly the set-valued reverse plane partitions whose reading word is a reverse lattice word, which yields the Schur expansion via the commuting relation $E_i(\mathrm{read}(T))=\mathrm{read}(e_i(T))$. For the omega image, the machinery is a family of Fomin--Greene type operators $\tilde u_i=u_i(1+d_{\mu,i})$ acting on partitions by adding vertical strips, whose noncommutative Cauchy identity converts the generating function into one over marked multiset-valued tableaux.

What would settle it

Search for a benign two-letter filling of some skew shape whose columns $j-1$ and $j$ are both descents, with column $j$ mixed and containing $\{1,2\}$, for which resolving $j$, then $j-1$, then $j$ gives a different filling from resolving $j-1$, then $j$, then $j-1$; such a pair would falsify Lemma 2.9 and break the well-definedness of the crystal operators, and it can be checked by exhaustive search over small shapes.

Watch

Extended reading notes

Core claim

For any skew shape $\lambda/\mu$, the hybrid Grothendieck polynomial $G_{\lambda/\mu}(\mathbf{x}_n;\mathbf{t};\mathbf{w})$ is a Schur-positive symmetric function. Its Schur expansion is $G_{\lambda/\mu}=\sum_{\gamma,\theta}\mathbf{t}^{\gamma}\mathbf{w}^{\theta}\sum_{\nu} H^{\nu,\gamma,\theta}_{\lambda/\mu,n} s_\nu$, where $H^{\nu,\gamma,\theta}_{\lambda/\mu,n}$ is the number of set-valued reverse plane partitions of weight $\nu$, column-equalities $\gamma$, and excess $\theta$ whose reading word is a reverse lattice word. The proof builds a $\mathfrak{gl}_n$-crystal on set-valued reverse plane partitions whose connected components are irreducible crystals of highest weight $\nu$, so each component contributes exactly one Schur function. As corollaries, straight-shape hybrid polynomials have saturated Newton polytopes, meaning every lattice point in the Newton polytope is an exponent vector, with degree given by the size of an explicit partition $\bar\lambda^{(n)}$, and the omega image of the one-parameter specialization is the generating function of marked multiset-valued tableaux.

Load-bearing premise

The entire construction of the crystal operators rests on the claim that a local rewriting procedure on two-letter fillings—flipping pure columns and then resolving descents—always reaches the same final filling regardless of the order in which the rewrites are applied; the one delicate overlapping case is verified only by 'it is easy to check', so if that overlap ever failed, the operators and the Schur expansion would collapse.

Editorial extensions

If this is right

  • The hybrid polynomial is Schur positive for every skew shape, so its specializations, including the refined stable and refined dual stable Grothendieck polynomials, all have coefficients counted by reverse-lattice-word fillings.
  • Every straight-shape hybrid Grothendieck polynomial has a saturated Newton polytope, meaning the support of the polynomial is exactly the lattice points of its convex hull, and the Newton polytope has the integer decomposition property.
  • The degree of $G_\lambda(\mathbf{x}_n;\mathbf{t};\mathbf{w})$ equals the size of an explicit partition $\bar\lambda^{(n)}$ built from the largest strict partition inside $\lambda$.
  • The omega image of $G_{\lambda/\mu}(\mathbf{x};\alpha;\beta)$ is a generating function over marked multiset-valued tableaux, and the two classical specializations recover the known omega images of stable and dual stable Grothendieck polynomials.
  • Because the crystal components are irreducible $\mathfrak{gl}_n$-crystals, the expansion coefficients are nonnegative integers and the same crystal provides a uniform combinatorial rule valid for all skew shapes at once.

Reading between the lines

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

  • Editorial extension: the same crystal may yield a uniform Littlewood–Richardson-type rule for the K-theoretic structure constants of Grassmannians at the hybrid level, interpolating between the known rules for the two specializations.
  • Editorial extension: the reverse-lattice-word criterion suggests a direct dynamic-programming algorithm for computing the coefficients $H^{\nu,\gamma,\theta}_{\lambda/\mu,n}$ by scanning rows, making the expansion computable for shapes beyond hand-drawn examples.
  • Editorial extension: the unproved confluence case in Lemma 2.9 is a natural target for computer-assisted checking; a counterexample there would break the crystal operators and the Schur expansion, while the symmetric-function identities might still survive through other means.
  • Editorial extension: the explicit interval $(\lambda_1)\subseteq\rho\subseteq\bar\lambda^{(n)}$ for straight shapes suggests a conjectural analogue for skew shapes, which the paper notes is open even for the classical stable and dual stable specializations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper defines hybrid Grothendieck polynomials G_{λ/μ}(x;t;w) as generating functions over set-valued reverse plane partitions (SVRPPs) of skew shape λ/μ, with statistics ircont, ceq, and ex. It specializes to refined stable Grothendieck polynomials (t=0) and refined dual stable Grothendieck polynomials (w=0), and to Schur functions when both t=w=0. The main results are: (1) symmetry in x, proved by an explicit involution constructed from a flip and descent-resolution algorithm on benign 12-SV tables; (2) a Schur expansion of G_{λ/μ}(x_n;t;w) with coefficients counted by SVRPPs whose reading word is a reverse lattice word, obtained by building a gl_n-crystal on SVRPP_n(λ/μ); (3) the saturated Newton polytope (SNP) property for straight shapes, via a characterization of possible weights of highest weight elements; and (4) a combinatorial formula for the omega-involution image in the specialization t_i=α, w_i=β, derived through Fomin–Greene noncommutative Schur operators. The paper also formulates several open problems and conjectures.

Significance. If the technical gaps are repaired, this paper would be a substantial unification: it places refined stable and refined dual stable Grothendieck polynomials into one crystal-theoretic framework and recovers or extends results of Monical–Pechenik–Scrimshaw, Galashin, Chan–Pflueger, and Galashin–Grinberg–Liu. The SNP result for straight shapes and the omega formula for the two-parameter specialization are new and likely to be useful. The paper is clearly written, with many worked examples and an appendix listing the full crystal graph for a small shape, which strengthens its accessibility. The main caveats are that several load-bearing local statements are asserted without detailed proof (the braid relation in Lemma 2.9, properties of L_i in Lemma 4.5, and the correctness of the reconstruction algorithm in Proposition 3.12), and the definitional framework for subtableaux T(i) with empty boxes needs to be made precise before the crystal operators are fully well-defined.

major comments (5)
  1. [§2.4, Lemma 2.9, Case 2] The braid identity res_{j−1}(res_j(res_{j−1}(T))) = res_j(res_{j−1}(res_j(T))) is asserted with only 'it is easy to check'. Because Proposition 2.10 (the uniqueness of the normal form) depends on this identity, and hence so do the well-definedness of the involution Φ in Theorem 2.2 and the crystal operators e_i/f_i in Section 3.2, a complete verification is required. The proof must cover all set-valued variants (the two starred entries in (M1) taking values {2} or {1,2}, and the two starred entries in (2M) taking values {1} or {1,2}) together with the benignity constraints on seplist.
  2. [§2.2, Definition 2.4] The 'extra constraint' in Definition 2.4 refers to 'there exists a (unique) box in column k_{j+1} filled with the set {1,2}', but a mixed column in a benign 12-SV table can contain more than one box filled with {1,2} (for example, the column {1}, {1,2}, {1,2}, {2} is weakly increasing). The uniqueness assertion is therefore not justified, and the definition is ambiguous. The authors should specify which box is meant (e.g., the topmost such box) and then re-verify the descent-resolution lemmas under that clarification.
  3. [§2.1 and §3.2] The subtableau T(i) of a SVRPP T obtained by keeping only the entries i and i+1 contains empty boxes at all other positions, but a 12-SV table is defined (Definition 1.1 and §2.1) as a filling of every box of λ/μ with a nonempty subset of {1,2}. Consequently the flip and descent-resolution operations are not literally applicable to T(i) or to the modified subfilling T^{(i)} used in the definition of e_i and f_i. The paper must either extend the entire framework of Sections 2 and 3 to allow empty boxes in 12-SV tables, together with all the relevant definitions of descents, benignity, and seplist, or justify that the boxes occupied by i and i+1 always form a skew shape in the cases needed. This is a load-bearing issue for Theorems 1.2 and 1.3.
  4. [§4, Lemma 4.5] The operator L_i is asserted to send SVRPP_n(λ) to itself and to preserve the property that the column reading word is a reverse lattice word, but no proof of either assertion is given. Since Lemma 4.5 is used to realize every partition ρ with (λ1) ⊆ ρ ⊆ ¯λ(n) as the weight of a highest weight element, the proof of Theorem 1.4 (SNP) depends on these properties. A detailed verification of the row and column inequalities under the operation L_i and of the reverse-lattice-word property of the column reading word of T* must be supplied.
  5. [§3.4, Proposition 3.12] The reconstruction algorithm in the proof of Proposition 3.12 is described informally, and its correctness is not proved. The argument should demonstrate that the procedure is deterministic and well-defined (e.g., that the 'rightmost box not filled' is always uniquely determined and that the case split on v_j > a always applies unambiguously), that it terminates, and that the resulting filling has the prescribed read(T), height(T), and ex(T). The uniqueness statement is used in Theorem 3.13 to obtain an injective embedding of crystal components into S_m^n, so the missing correctness proof is load-bearing for Theorem 1.3.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'polyotpe' for 'polytope', 'coutable' for 'countable', 'crytal' and 'crytal graph' for 'crystal' and 'crystal graph', 'Grothedieck' for 'Grothendieck', 'semi-nomal' for 'seminormal', 'unmarkd content' for 'unmarked content', 'formuals' for 'formulas', 'Integeral lattice model' for 'Integrable lattice model', and 'theirin' for 'therein'.
  2. [§2.2, Definition 2.4] The phrase 'some (unique) box' also appears later in the paper, but uniqueness of a box containing a particular set in a column is not generally a consequence of the defining inequalities; each such use should be checked.
  3. [§5] The definition of marked multiset-valued tableaux in Definition 5.7 is clear in examples, but the marking condition 'if there exists an i located at a higher position in the same column' should be stated more precisely for boxes that contain multiple copies of i, since the row-order convention for permuting elements inside a box needs to be explicit.
  4. [References] The reference list is extensive, but [15] is a self-citation that is not directly connected to the central arguments; the authors may wish to indicate its relevance in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are derived from explicit combinatorial definitions and external standard results; the main soft spot is an unverified local braid computation, not a circular dependency.

full rationale

The paper's derivation chain is self-contained and non-circular. The hybrid Grothendieck polynomial is defined directly as a weight generating function over SVRPPs (eq. (1)), and Theorem 1.2's symmetry is proved by an explicitly constructed involution (flip followed by descent resolutions), not by assuming symmetry. Theorem 1.3's Schur expansion is obtained by building a crystal on SVRPP_n(λ/μ), proving a seminormal crystal structure from the definitions of the operators (Prop. 3.3, Cor. 3.5), embedding each component into the word crystal via the reading word (Props. 3.12, 3.17, Cor. 3.18), and then invoking the external fact that word crystals have Schur characters ([19], Prop. 3.1); the reverse-lattice-word description of the coefficients follows from Proposition 3.8, a characterization proved from the crystal operators. Theorem 1.4's SNP result uses the external criterion of [48] plus interval statements proved directly from T_min/T_max and the highest-weight characterization. Theorem 1.5's omega formula uses the external Fomin-Greene noncommutative Cauchy identity ([18], Thm. 5.2) after proving the operator relations in Theorem 5.3. None of these steps assumes the target polynomial, its symmetry, or its Schur expansion as an input. The only self-citation, [15], appears in a 'see for example' list in Section 6.4 and is not load-bearing. One verification gap should be noted: Lemma 2.9's Case 2 asserts the braid identity with 'it is easy to check' (Section 2.4), and this local relation is indeed load-bearing for the well-definedness of Φ and hence of the crystal operators. However, that is an omitted case check, not a circular reduction: it does not assume the target result, so it does not affect the circularity score.

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

No free parameters are fitted. The results are proved from explicit definitions and external standard theorems. The only new combinatorial objects, SVRPPs and marked multiset-valued tableaux, are precisely defined rather than postulated entities, so no invented entities are charged.

assumptions (4)
  • standard math Normal gl_n-crystals have characters equal to Schur polynomials for each connected component (Kashiwara crystal basis theory).
    Invoked in Section 3.5 to identify components of the crystal on SVRPP_n with B(ν) and to compute characters.
  • standard math Fomin-Greene Theorem 5.2: operators satisfying relations (i)-(iii) satisfy Cauchy-type identities; the noncommutative Schur functions are well-defined.
    Used in Section 5 to compute the omega image via A(x) and B(x) products.
  • standard math SNP criterion of Nguyen et al. (Theorem 4.1): a symmetric polynomial that is a signed combination of Schur polynomials with interval support μ⊆ρ⊆ν has saturated Newton polytope.
    Used in Section 4 as the external criterion for Theorem 1.4.
  • standard math The descent-resolution operations of Galashin-Grinberg-Liu [20] for reverse plane partitions are correct; the new set-valued variants reduce to them when the entries are singleton sets.
    Remark 2.3 states the reduction; the proof of Lemma 2.9 relies on the corresponding local checks in [20].

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Pith. "Pith review of Hybrid Grothendieck polynomials." pith.science (2026). https://pith.science/paper/6KSVSC7K

@misc{pith2026250519072,
  author       = {Pith},
  title        = {Pith review of: Hybrid Grothendieck polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KSVSC7K}},
  note         = {Machine review of arXiv:2505.19072}
}
abstract

For a skew shape $\lambda/\mu$, we define the hybrid Grothendieck polynomial $${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w}) =\sum_{T\in \mathrm{SVRPP}(\lambda/\mu)} \textbf{x}^{\mathrm{ircont}(T)}\textbf{t}^{\mathrm{ceq} (T)}\textbf{w}^{\mathrm{ex}(T)}$$ as a weight generating function over set-valued reverse plane partitions of shape $\lambda/\mu$. It specializes to \begin{itemize} \item[(1)] the refined stable Grothendieck polynomial introduced by Chan--Pflueger by setting all $t_i=0$; \item[(2)] the refined dual stable Grothendieck polynomial introduced by Galashin--Grinberg--Liu by setting all $w_i=0$. \end{itemize} We show that ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ is symmetric in the $\textbf{x}$ variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ in the basis of Schur functions, extending previous work by Monical--Pechenik--Scrimshaw and Galashin. Based on the Schur expansion, we deduce that hybrid Grothendieck polynomials of straight shapes have saturated Newton polytopes. Finally, using Fomin--Greene's theory on noncommutative Schur functions, we give a combinatorial formula for the image of ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ (in the case $t_i=\alpha$ and $w_i=\beta$) under the omega involution on symmetric functions. The formula unifies the structures of weak set-valued tableaux and valued-set tableaux introduced by Lam--Pylyavskyy. Several problems and conjectures are motivated and discussed.

Figures

Figures reproduced from arXiv: 2505.19072 by the authors.

Figure 1
Figure 1. T1, T2 and T3 are SVRPP’s. Furthermore, T2 is a set-valued tableau, and T3 is a reverse plane partition. Here we have omitted the braces { } for sets as well as the commas between numbers. We use SVRPP(λ/µ) to denote the set of all SVRPP’s of shape λ/µ. Write (i, j) for the box in row i and column j in the matrix coordinate, and for T ∈ SVRPP(λ/µ), let T(i, j) be the set filled in (i, j). Define • irredundant conten… view at source ↗
Figure 2
Figure 2. Specializations of hybrid Grothendieck polynomials. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The Newton polytope of G(4,2,1)(x2; 1; 1). To prove Theorem 1.4, we apply Theorem 1.3 to show that for any given λ, there exist partitions µ ⊆ ν such that sρ(xn) appears in the expansion of Gλ(xn; t; w) if and only if µ ⊆ ρ ⊆ ν. This, along with a criterion for SNP given in [48], leads to a proof of Theorem 1.4. As a byproduct, we get a combinatorial description for the degree of Gλ(xn; t; w), see Corollary 4.8. Fin… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The descent-resolution operations. Remark 2.3 When neither ∗ nor ⋆ is equal to {1, 2}, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Examples of the descent-resolution operations. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Illustration of the map Φ. 2.3 Descent resolution Let T ∈ BSVT12(λ/µ). Assume that k is a descent of T. Then the k-th column of T is either 2-pure or mixed, and the (k + 1)-th column of T is either 1-pure or mixed. Note that the k-th and the (k +1)-th columns of T cann…
Figure 7
Figure 7. Figure 7: T1 is benign and T2 is not benign. (3) We have ℓ(T) > ℓ(resk(T)). Here ℓ(T) = X j≥1 j · sig(the j-th column of T), where, for a column C of T, sig(C) =    0, if C is empty or 2-pure, 1, if C is mixed, 2, if C is 1-pure. Proof. The assertions in (1), (2) and (3) in…
Figure 8
Figure 8. Figure 8: Illustration of the crystal operator ei with i = 1. (3) If ei(T) = T ′ , then wt(T ′ ) = wt(T) + ei − ei+1. Proof. (1) This follows from the definitions of ei and fi . (2) Let first check that ei(T) = T ′ implies T = fi(T ′ ). As before, use T (i) to denote the subtabl…
Figure 9
Figure 9. Figure 9: Illustration of crystal operator fi with i = 2. Let Di be the column where the descent-resolution procedure stops. Clearly, we have Ci < Di ≤ Ai . By the above analysis, Di is exactly the column corresponding to the rightmost uncanceled + in (T ′ ) (i) . To construct f…
Figure 10
Figure 10. Figure 10: Two of the connected components in the crystal graph of SVRPP [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: read(T) = 64 · 5 | 75 · 6 | 432 · 13 | 762 · 1345. Remark 3.7 The reading word of a SVRPP incorporates the constructions of the reading word of a set-valued tableau and the reading word of a reverse plane parti￾tion defined in [5]. When restricting to reverse plane pa…
Figure 12
Figure 12. Figure 12: Illustration of the distribution of i and i + 1 in (2.3). We now give a proof of Proposition 3.8. Proof of Proposition 3.8. Assume that T is a highest weight element, that is, ei(T) = 0 for every i ∈ [n − 1]. This means that all (i + 1)-pure columns of T are paired wi…
Figure 13
Figure 13. Figure 13: Tmax for λ = (7, 5, 4, 4, 1, 1) and n = 3, 7, respectively. Lemma 4.4 For any partition λ and positive integer n, Tmax is a highest weight element in SVRPPn (λ) with wt(Tmax) = λ¯(n) . Proof. We first check that Tmax has weight equal to λ¯(n) . Write wt(Tmax) = (c1, .…
Figure 14
Figure 14. Figure 14: Illustration of operators ui and dµ,i. Consider the operator u˜i = ui(1 + dµ,i). Set A˜(x) = · · ·(1 + xu˜3)(1 + xu˜2)(1 + xu˜1). Proposition 5.1 The coefficient of λ in the product · · · A˜(x3)A˜(x2)A˜(x1) µ is equal to Gλ/µ(x; α; β). 29 [PITH_FULL_IMAGE:figures/ful…
Figure 15
Figure 15. Figure 15: A marked multiset-valued tableau. Remark 5.8 In the case when mark(T) = 0 (resp., ex(T) = 0), a marked multiset￾valued tableau T becomes a weak set-valued tableau (resp., a valued-set tableau), as defined by Lam and Pylyavskyy [34]. Let MMSVT(λ/µ) be the set of marked…

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  1. A Lam--Postnikov--Pylyavskyy inequality for hybrid Grothendieck polynomials

    math.CO 2026-07 accept novelty 6.5 of 10

    Hybrid Grothendieck polynomials satisfy a multivariate LPP Schur-positivity inequality that unifies and refines the stable and dual-stable cases.

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Pith tools

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