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Skew Hives, Skew Skeps, Skew Schur Log-Concavity

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A new proof with skew hives and skew skeps shows that when paired skew shapes lie in a common L-convex hull, the corresponding product of skew Schur functions dominates the other product in Schur-positive order.

desk verdict A genuinely new skew Schur log-concavity result built on Speyer's machinery, but the key section S1/S2 proof is sketched rather than verified. read the letter →

arxiv 2608.13544 v1 pith:3T3VU46W submitted 2026-08-13 math.CO

Tuong Le , Son Nguyen This is my paper · ORCID
classification math.CO MSC 05E0505E1005A17
keywords skewhivesskepsSchurlog-concavityLittlewood-RichardsoncoefficientsL-convexityoctahedronrecurrenceNewell-Littlewoodnumbersshadowfunctions
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

Skew Schur functions are symmetric functions indexed by a difference of two partitions; their products have nonnegative Littlewood-Richardson coefficients, but it is usually hard to tell when one product dominates another. This paper introduces two counting models, skew hives and skew skeps, for the coefficient $c^\kappa_{\lambda/\mu,\nu/\rho}$, and uses them to prove a clean dominance statement: whenever two pairs of skew shapes have the same coordinate-wise sum and the second pair lies in the L-convex hull of the first, the product for the second pair minus the product for the first is Schur positive. That result is the skew analog of the classical Schur log-concavity conjecture, and it directly gives log-concavity inequalities for Newell-Littlewood numbers and shadow skew Schur functions. The same models are connected by explicit bijections to peelable tableaux, yielding several equivalent Littlewood-Richardson rules.

What carries the argument

The central object is the skew skep: a function on the triangular board $\Delta_{2n}$ satisfying four rhombus inequalities, with upper and right boundary data $(\lambda_1,\nu_1,\ldots,\lambda_n,\nu_n)$ and $(\mu_1,\rho_1,\ldots,\mu_n,\rho_n)$ and northwest boundary $\kappa$; Theorem 2.18 says the number of such skew skeps equals $c^\kappa_{\lambda/\mu,\nu/\rho}$. The skew hive is the sibling model with the three hive inequalities, imposed except along one diagonal; Theorem 2.13 gives the same count. The two models are connected by the octahedron recurrence on a three-dimensional point set $\Pi_n$: a solution's restrictions to four different sections are respectively skew hives and skew skeps with interchanged boundaries, and Theorem 2.23 says all four restrictions are valid exactly when the solution satisfies every rhombus inequality. The proof then steps outside the models: fixing the even-parity entries and $\kappa$ makes the set of skew skeps an L-convex set in the sense of discrete convex analysis, and the marginal-count theorem for L-convex sets makes the boundary count L-log-concave.

What would settle it

For $n=4$, enumerate all integer functions on $\Pi_4$ satisfying the octahedron recurrence and test whether every one whose bottom section is a valid skew hive also satisfies every rhombus inequality; one failure disproves Theorem 2.23. Independently, compute $c^\kappa_{\lambda'/\mu',\nu'/\rho'}-c^\kappa_{\lambda/\mu,\nu/\rho}$ for all $\kappa$ for a small admissible quadruple of partitions with four rows; any negative value disproves Theorem 3.12.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.12, is that for partitions $\lambda,\mu,\nu,\rho$ and $\lambda',\mu',\nu',\rho'$ satisfying $(\lambda,\mu)+(\nu,\rho)=(\lambda',\mu')+(\nu',\rho')$ and $(\lambda',\mu'),(\nu',\rho')\in L((\lambda,\mu),(\nu,\rho))$ in $\mathbb{Z}^{2n}$, the difference $s_{\lambda'/\mu'}s_{\nu'/\rho'}-s_{\lambda/\mu}s_{\nu/\rho}$ is Schur positive. Phrased in the paper's model, the coefficient of $s_\kappa$ in this difference is a difference of two skew-skep counts, and the proof shows the primed count is always at least the unprimed one. The mechanism is that after fixing the parity sublattice of the triangular board and the northwest boundary $\kappa$, the set of skew skeps carrying those fixed data is an L-convex set; the marginal theorem for L-convex sets makes its boundary count an L-log-concave function of the boundary pair, and a reflection symmetry of the skew-skep model converts that inequality into the desired coefficient-wise dominance.

Load-bearing premise

The proof of the octahedron equivalence depends on two geometric slices through a three-dimensional grid that are exhibited only in figures, with the authors saying the reader can check them; if those slices do not cut every required plane correctly, the skew-skep counting theorem and the main log-concavity proof collapse.

Editorial extensions

If this is right

  • In the non-skew case ($\mu=\rho=\mu'=\rho'=\varnothing$), Theorem 3.12 reduces to the classical Schur log-concavity statement: $s_{\lambda'}s_{\nu'}-s_\lambda s_\nu\ge 0$ in Schur-positive order whenever $\lambda'+\nu'=\lambda+\nu$ and the primed partitions lie in the $L^\natural$-hull.
  • The log-concavity passes to Newell-Littlewood numbers: $N_{\mu',\nu',\lambda}\ge N_{\mu,\nu,\lambda}$ for every $\lambda$ when $\mu+\nu=\mu'+\nu'$ and $\mu',\nu'$ lie in the $L^\natural$-hull; equivalently, the same positivity holds in the universal character basis for symplectic and orthogonal groups.
  • Shadow skew Schur functions satisfy the same theorem: $S^{(k,\ell)}_{\lambda'/\mu'}S^{(k,\ell)}_{\nu'/\rho'}-S^{(k,\ell)}_{\lambda/\mu}S^{(k,\ell)}_{\nu/\rho}$ is Schur positive under the same hypotheses.
  • The earlier min/max, integer-averaging, and sorting inequalities for products of skew Schur functions all follow as special cases of Theorem 3.12.
  • Skew hives, skew skeps, and phased peelable tableaux give $2n$ equivalent Littlewood-Richardson rules, related by the octahedron recurrence and Bender-Knuth involutions.

Reading between the lines

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

  • Beyond the paper, the same L-convex marginal argument should extend to products of more than two skew Schur functions if a multi-boundary skew-skep model can be written down; the final remarks only gesture at higher-dimensional path models.
  • The coefficient-wise nature of the proof suggests there should be an explicit injective map from the skew skeps counted by $c^\kappa_{\lambda/\mu,\nu/\rho}$ into those counted by $c^\kappa_{\lambda'/\mu',\nu'/\rho'}$; constructing such a map would turn the Schur-positive difference into a bijective proof with extra structure.
  • Because Theorem 1.6 is proved by the same skew-skep count with only the first $2\ell$ boundary entries fixed, the shadow-Schur argument likely works verbatim for lower Schur functions defined by fixing bottom parts, which the authors list as an open question.
  • The sections $S_1,S_2$ that the proof leaves to figures could be produced algorithmically for each $n$; a finite search for triangulated sections with the stated transversality would replace the pictorial check with a computation.
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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

2 major / 6 minor

Summary. The paper introduces two new combinatorial models, skew hives and skew skeps, on the triangular grid Δ_{2n}, and proves that both count Littlewood–Richardson coefficients c^κ_{λ/μ,ν/ρ} for products of skew Schur functions. The main structural result is an octahedron-recurrence equivalence between four sections (Theorem 2.23), from which the skew skep counting theorem (Theorem 2.18) is derived. Using Speyer's L-convexity machinery and the symmetry between SkewSkep^κ_{λ/μ,ν/ρ} and SkewSkep^κ_{ν/ρ,λ/μ}, the paper proves Theorem 3.12, a skew generalization of the Lam–Postnikov–Pylyavskyy Schur log-concavity conjecture, with applications to Newell–Littlewood numbers and shadow skew Schur functions. Section 5 gives bijections between phased skew hives and phased peelable tableaux, answering a question of Speyer. The counting and log-concavity arguments are supported by worked examples, but the proof of Theorem 2.23 contains a substantial unverified geometric step.

Significance. If fully substantiated, the main theorem is a genuine advance: it generalizes Speyer's resolution of the Lam–Postnikov–Pylyavskyy conjecture to skew Schur functions and simultaneously recovers several results of Lam–Postnikov–Pylyavskyy. The skew hive and skew skep models are natural and likely to be useful tools, and the applications to Newell–Littlewood numbers and shadow Schur functions are nontrivial. Section 5 provides a conceptual bridge between peelable tableaux and octahedron-recurrence models. The paper is clearly written and the main logical chain is visible, but the proof of Theorem 2.23 currently leaves a load-bearing transversality and coverage verification to the reader, so the central claim is not yet fully established.

major comments (2)
  1. [§2.4, Theorem 2.23] The proof of the direction (3)⇒(5), and similarly (2)⇒(5), is not a verification. Definition 2.25 explicitly cautions that a section is not determined by its discrete set of integer points, yet the sections S1 and S2 are introduced only through the t-coordinate pictures in Figures 2b and 2c, with the sentence 'We leave the explicit descriptions to the readers' and the assertion 'One can check that S1 intersects wavefronts of type (1) and (2) transversely...'. Lemma 2.30 requires, for every wavefront W, a section S(W) transverse to W along which all rhombus inequalities hold; Figures 4 and 5 illustrate the intended intersections but do not specify the triangulations needed to check transversality or coverage. Since Theorem 2.18 depends on Theorem 2.23, and Theorem 3.12 uses the equality |SkewSkep^κ_{λ/μ,ν/ρ}(g+)| = |SkewSkep^κ_{ν/ρ,λ/μ}(g+)| obtained from Theorem 2.23 to take square roots in equation (3), this gap is load-bearing. Please provide explicit descriptions of S1 and S2 and a complete proof of the transversality and coverage properties, or replace Lemma 2.30 with a fully verified statement.
  2. [§5.3, Bijection 5.14] The proof of Bijection 5.14 is incomplete in its induction step. After the base case k=n, the text asserts that passing from k to k−1 is achieved by the octahedron recurrence at diagonals i+j=k,k+2,…,2n−k on the hive side and by BK_k∘BK_{k+2}∘⋯∘BK_{2n−k} on the tableau side, and then states 'It is not difficult to check that Bender–Knuth involutions coincide with the octahedron recurrence after the procedure.' This coincidence is exactly the content of the bijection and is not a routine check; it needs a proof. The worked Example 5.15 illustrates one instance (k=3 to k=2) but does not establish the general claim. As it stands, Theorem 5.13 and the claimed answer to Speyer's question are not fully verified.
minor comments (6)
  1. [§3.1, Definition 3.1] The displayed definition of L(u,v) reads 'min(...) ≤ z_i−z_j ≤ min(...)' with the same expression on both sides; the upper bound should be max(u_i−u_j, v_i−v_j). This appears to be a typo, but it makes the definition formally nonsense as printed.
  2. [Theorem 3.12] The equation in the theorem statement, '(λ, μ) + (ν, ρ) = (λ′, µ′) + (µ′, ρ′)', appears to contain a typo: the second summand on the right should be (ν′, ρ′) rather than (µ′, ρ′).
  3. [§1.1] The notation ≤_s for 'Schur-positive difference' is used in the introduction and in Theorem 3.12 but is never explicitly defined; please add a definition at first use.
  4. [§1.2] In the sentence 'Speyer's proof techinique shows...' there is a typo: 'techinique' should be 'technique'.
  5. [§2.2, Theorem 2.13] The proof of Theorem 2.13 verifies the boundary h↖ by writing '= h↑ = κ'; this is correct only because h↖ of the constructed skew hive equals the h↑ boundary of the original hive. The notation is potentially confusing and would benefit from an explicit sentence saying which boundary of which object is being used.
  6. [§5.3] The definition of k-phased skew hive is given by rhombus inequalities determined by Ht_{P_k}, but the explicit inequalities are not written down; for a self-contained paper, at least the k=2 case (used in Example 5.15) should be stated.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the main skew Schur log-concavity theorem is derived from the new skew skep model plus Speyer’s external L-convexity results, and the only overlapping-author citations occur in Section 5, where they are not load-bearing.

full rationale

The central derivation chain is not circular. Theorem 3.12 is proved by counting skew Schur Littlewood–Richardson coefficients with skew skeps (Theorem 2.18), applying Speyer’s external L-convex marginal theorem (Theorem 3.8), and using the octahedron-recurrence symmetry supplied by Theorem 2.23 to pass from a product bound to a square root. The statement being proved, skew Schur log-concavity, is not assumed in any of these ingredients. The skew hive and skew skep models are new objects, and their counting theorems are proved from the ordinary hive description of Littlewood–Richardson coefficients together with explicit bijections, not from the target Schur positivity statement. The only overlapping-author citations are to [NNW25], and they appear in Section 5, where peelable tableaux are connected to skew hives; those bijections are independent of the main log-concavity claim and are not needed for Theorem 3.12, so they are not load-bearing. The most notable weakness is not circularity but completeness: the proof of Theorem 2.23 delegates the existence and transversality of the auxiliary sections S1 and S2 to Figures 2b, 2c, 4, and 5, with the text “We leave the explicit descriptions to the readers” and “One can check”. This is an omitted verification that could affect Theorem 2.18 and therefore the square-root step in Theorem 3.12, but it is a proof gap, not a reduction of the conclusion to its own inputs. Against external benchmarks (Speyer’s theorems, hives, and the classical Littlewood–Richardson rule), the paper’s main derivation is self-contained.

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

The paper introduces new combinatorial models (skew hives, skew skeps, k-phased skew hives) but these are definitions within the proof, not postulated entities requiring independent evidence. The main theorem is derived from external results in discrete convex analysis (Speyer, Murota, Henriques-Kamnitzer) and from the newly proven skew skep counting theorem. No free parameters are fitted. The most fragile assumed input is the correctness of Speyer's L-convexity theorems and the unproven geometric sections S1 and S2.

assumptions (4)
  • standard math Speyer's Theorem 3.8: for an L-convex set K, the fiber count E_K(x) is L-log-concave.
    Invoked in the proof of Theorem 3.12 to obtain log-concavity of the skew skep fiber counts; cited from [Spe26] and not proved here.
  • standard math The octahedron recurrence section-propagation result [HK06, Lemma 3.1] used to transfer rhombus inequalities between sections.
    Used in Lemma 2.30, following [Spe26, Lemma 3.26]; the proof relies on the cited lemma.
  • domain assumption The identification L(u,v) = P_{u,v} between the L-convex hull and the minimal alcoved parallelepiped containing u and v.
    Stated in Section 3.1 without proof or citation; this identification connects the main theorem to the original parallelepiped formulation of Conjecture 1.1.
  • standard math Murota's discrete convex analysis theorems, in particular the equivalence of Speyer's L-convexity definitions with Murota's (Theorems 4.3 and 4.4 of [Spe26]).
    Used implicitly in Definitions 3.2 and 3.3 and in the application of Theorem 3.8.

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Pith. "Pith review of Skew Hives, Skew Skeps, Skew Schur Log-Concavity." pith.science (2026). https://pith.science/paper/3T3VU46W

@misc{pith2026260813544,
  author       = {Pith},
  title        = {Pith review of: Skew Hives, Skew Skeps, Skew Schur Log-Concavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3T3VU46W}},
  note         = {Machine review of arXiv:2608.13544}
}
read the original abstract

Knutson and Tao's hives is a combinatorial model to compute Littlewood--Richardson coefficients. Similar to hives, Speyer introduced skeps and used them to prove a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy. We first introduce skew hive and skew skep models, which specialize to both hives and skeps, and use this to prove a skew Schur log-concavity result generalizing Lam--Postnikov--Pylyavskyy conjecture. As a consequence, we obtain some log-concavity results concerning Newell--Littlewood numbers and shadow skew Schur functions. Finally, we explain bijections between (skew) hives, (skew) skeps, and peelable tableaux by Nguyen--Nguyen--Woodruff, answering Speyer's question.

Figures

Figures reproduced from arXiv: 2608.13544 by the authors.

Figure 1
Figure 1. shows the t-coordinates of S bot skewhive and S bot skewskep in Πn. The black bold edges are short edges of the green rhombi. The readers can check that the rhombi inequalities in Figure 1a (resp. Figure 1b) match the inequality in Definition 2.12 (resp. Definition 2.17). 0 −1 −1 −2 −2 −2 −3 −3 −3 −3 −2 −2 −2 −2 −2 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 (a) Skew hive 0 −1 −1 0 0 0 −1 −1 −1 −1 0 0 0 0 0 −1 −1 −1 −1 −1 −1 0 … view at source ↗
Figure 2
Figure 2. Sections Definition 2.27. A wavefront W is a two-dimensional subset of RΠn of one of these types: (1) The intersection of RΠn with {t − c = i − j}, for c even, |c| < 2n. (2) The intersection of RΠn with {t − c = j − i}, for c even, |c| < 2n. (3) The intersection of RΠn with {|t − c| = i + j}, for c even, 0 < |c| < 2n. We say that a wavefront W and a section S intersect transversely if W ∩ S has dimension ≤ 1 and in … view at source ↗
Figure 3
Figure 3. Intersection of Sbot with the wavefronts Proof. This is implicit in [HK06] and [Spe26]. The only caveat is that for wavefronts of type (3), we do not use the wavefront {|t| = i + j}. This is because this wavefront does not intersect the interior of RΠn and hence is not relevant. Lemma 2.30. Let h˜ : Πn → Z obeying the octahedron recurrence. Suppose for every wavefront W, there is a section S(W) transverse to W such … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Intersection of S1 with the wavefronts of types (1) and (2) 0 −1 −1 0 0 0 −1 −1 −1 −1 0 0 0 0 0 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 (a) Type (3) positive c 0 −1 −1 0 0 0 −1 −1 −1 −1 0 0 0 0 0 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 (b) Type (3) negative c [PITH_FULL_IMAGE:figures…
Figure 5
Figure 5. Figure 5: Intersection of S2 with the wavefronts of type (3) and h → 1 = (ρ, µ) ⇐⇒ h → 2 = (ρ1, µ1, . . . , ρn, µn) ⇐⇒ h → 3 = (µ1, ρ1, . . . , µn, ρn) ⇐⇒ h → 4 = (µ, ρ). 3 Skew Schur log-concavity 3.1 L-convexity Murota [Mur03] developed two dual notions of discrete convexity, …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.