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arxiv: 2606.06688 · v1 · pith:IPGQ7LT4new · submitted 2026-06-04 · 🧮 math.CO

Correlation inequalities for Schur positivity

Swee Hong Chan , Hong Chen , Igor Pak , Daniel Soskin This is my paper

Pith reviewed 2026-06-28 00:09 UTC · model grok-4.3

classification 🧮 math.CO
keywords Schur positivityAhlswede-Daykin inequalitystable Grothendieck polynomialslog-supermodularitycorrelation inequalitiesposet inequalities
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The pith

Schur-positive functions obey a generalized version of the Ahlswede-Daykin inequality.

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

The paper extends the 1978 Ahlswede-Daykin inequality on correlation to the setting of Schur-positive functions. This new inequality also recovers the 2007 Lam-Postnikov-Pylyavskyy inequality as a special case. The authors apply the result to settle Mihalcea's conjecture that stable Grothendieck polynomials are log-supermodular. A reader might care because these inequalities control the signs of differences in combinatorial counts and appear in the study of positivity in algebraic combinatorics.

Core claim

The Ahlswede-Daykin inequality generalizes to a Schur positive ADS inequality for Schur-positive functions, which contains the Lam-Postnikov-Pylyavskyy inequality as a special case. This generalization resolves Mihalcea's conjecture on the log-supermodularity of stable Grothendieck polynomials.

What carries the argument

The Schur positive ADS inequality, which extends the classical ADS inequality to functions that are Schur-positive.

If this is right

  • The inequality applies directly to stable Grothendieck polynomials, proving they are log-supermodular.
  • Further generalizations of the inequality are possible under the same Schur-positivity conditions.
  • Applications to other Schur-positive combinatorial objects follow from the main result.

Where Pith is reading between the lines

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

  • If the inequality holds more broadly, it could apply to other positivity notions like Schubert positivity.
  • Testing the inequality on explicit small examples of Schur-positive functions would provide immediate verification.
  • Connections to other correlation inequalities in probability and combinatorics may emerge from this extension.

Load-bearing premise

The Schur-positive functions satisfy the same structural conditions as those required for the original Ahlswede-Daykin inequality to hold.

What would settle it

A concrete Schur-positive function on a poset that violates the ADS inequality would disprove the generalization.

Figures

Figures reproduced from arXiv: 2606.06688 by Daniel Soskin, Hong Chen, Igor Pak, Swee Hong Chan.

Figure 6.1
Figure 6.1. Figure 6.1: Multiplicative generators for TL4(2). 6.2. Temperley–Lieb immanants. Following Littlewood [Lit40] and Stanley [Sta00], the im￾manants are polynomials in matrix entries defined as follows. Fix k ≥ 1 and let Sk be symmetric group on k elements. Given a function f : Sk → C and a k × k matrix X = (xij )1≤i,j≤k, the f-immanant of X is the polynomial (6.2) Immf (X) := X w∈Sk f(w) x1,w1 · · · xk,wk . From now o… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Remaining basis elements in B4. where si = (i, i+ 1) is an adjacent transposition. Since (κ)κ∈Bk forms a basis of TLk(2), for each κ ∈ Bk, define a function fκ : Sk → R as σ(w) := X κ∈Bk (6.4) fκ(w) κ. Following [RS05, RS06], the Temperley–Lieb immanant of a matrix X, denoted Immκ(X), is defined as (6.5) Immκ(X) := Immfκ (X) = X w∈Sk fκ(w) x1,w1 · · · xk,wk . 6.3. From diagrams to multisets. Fix ℓ ≥ 1 an… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Diagrams compatible with (M, N ) in Example 6.1. The top and bottom rows are labeled by the elements of M and N , respectively. White nodes indicate elements in ψ(K). 2 4 7 8 10 11 1 2 3 3 5 6 2 4 7 8 10 11 1 2 3 3 5 6 [PITH_FULL_IMAGE:figures/full_fig_p017_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: 2 4 7 8 10 11 1 2 3 3 5 6 [PITH_FULL_IMAGE:figures/full_fig_p017_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Two compatible diagrams in Θ(Mλ⊔Nρ) as in Example 6.6. Vertices are labeled by λ ∗ i , µ∗ i , ν∗ i , and ρ ∗ i , where λ ∗ i = λi + ℓ + 1 − i (with µ ∗ i , ν∗ i , and ρ ∗ i defined analogously). These diagrams are equivalent to those in [PITH_FULL_IMAGE:figures/full_fig_p019_6_5.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: White nodes indicate elements in Ψ(E ⊔ F) = {u1, . . . , u6} = {1, 5, 6, 7, 8, 9}. Proof. For any i ∈ [ℓ], we have pi ∈ Ψ(E ⊔ F) ⇐⇒ ℓ − i + 1 ∈ E ⇐⇒ αℓ−i+1 + γℓ−i+1 + i ∈ Mλ(E) ⇐⇒ ψ(αℓ−i+1 + γℓ−i+1 + i) ∈ ψ [PITH_FULL_IMAGE:figures/full_fig_p022_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: (a) A diagram κ that is compatible with E ⊔ F, where E, F are as in Example 7.12. White nodes indicate elements in Ψ(E ⊔ F). (b) the graph Gκ, where the additional edges are colored in green. Lemma 7.13. Let κ ∈ B2ℓ be a Temperley–Lieb diagram compatible with E ⊔F, and let E ⊔F ⊆ E ⊔ F be an admissible set. Then κ is compatible with E ⊔ F if and only if the set E ⊔ F is a union of equivalence classes of … view at source ↗
read the original abstract

We generalize the Ahlswede--Daykin inequality (1978) to a Schur positive \emph{ADS inequality}, which also contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

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

0 major / 2 minor

Summary. The manuscript generalizes the Ahlswede--Daykin inequality (1978) to a Schur-positive ADS inequality that contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. It presents further generalizations and applications, and resolves Mihalcea's conjecture on the log-supermodularity of stable Grothendieck polynomials.

Significance. If the central claims hold, the work would meaningfully extend classical correlation inequalities into the setting of Schur positivity, offering a unified framework that recovers known results as special cases while settling an open conjecture. The combinatorial approach could facilitate new applications in algebraic combinatorics and symmetric function theory.

minor comments (2)
  1. The abstract states the generalization holds under the conditions of the classical ADS inequality, but the manuscript should explicitly restate those conditions in §1 or §2 to make the scope of the Schur-positive extension fully transparent without requiring the reader to consult the 1978 reference.
  2. Notation for the stable Grothendieck polynomials and the log-supermodularity statement in the resolution of Mihalcea's conjecture should be introduced with a brief self-contained definition in the section where the conjecture is addressed, even if standard in the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the main contributions: the Schur-positive generalization of the Ahlswede--Daykin inequality (containing the Lam--Postnikov--Pylyavskyy inequality as a special case) and the resolution of Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained generalization

full rationale

The paper claims a generalization of the external Ahlswede--Daykin inequality (1978) to Schur-positive functions (containing the external Lam--Postnikov--Pylyavskyy 2007 result as a special case) and a resolution of Mihalcea's conjecture. No equations, definitions, or steps in the abstract reduce by construction to fitted inputs or self-citations. All cited results are from independent prior work with no author overlap indicated. The central claims rest on combinatorial arguments whose validity is independent of the present paper's inputs, yielding a normal non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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