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arxiv: 2601.05007 · v2 · submitted 2026-01-08 · 🧮 math.CO

L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy

David E Speyer This is my paper

Pith reviewed 2026-05-16 16:01 UTC · model grok-4.3

classification 🧮 math.CO
keywords Schur functionsLittlewood-Richardson coefficientsL-log-concavityskepspartitionsnonnegative coefficientsL-convexity
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The pith

Under equal partition sums and bounded differences, the difference of Schur function products is Schur nonnegative.

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

This paper proves the conjecture of Lam, Postnikov and Pylyavskyy. It shows that if partitions λ, μ, λ', μ' satisfy λ + μ = λ' + μ' and the row differences of λ' lie between the minimum and maximum of those from λ and μ, then s_λ' s_μ' minus s_λ s_μ expands with nonnegative coefficients in the Schur basis. The argument introduces skeps as a combinatorial count of the Littlewood-Richardson coefficients appearing in the product and establishes that these counts obey L-log-concavity. The result follows because L-log-concavity on the skeps transfers directly to the desired Schur nonnegativity. A reader cares because the conditions give a concrete, checkable criterion for comparing products of Schur functions.

Core claim

The paper proves that for partitions λ, μ, λ', μ' with λ + μ = λ' + μ' and min(λ_i − λ_j, μ_i − μ_j) ≤ λ'_i − λ'_j ≤ max(λ_i − λ_j, μ_i − μ_j) for all i < j, the difference s_λ' s_μ' − s_λ s_μ is a nonnegative linear combination of Schur functions. The proof defines skeps, a new model for the Littlewood-Richardson coefficients in the product of two Schur functions, and applies Murota's L-convexity theory to obtain L-log-concavity for the skep counts.

What carries the argument

Skeps, a combinatorial object that enumerates Littlewood-Richardson coefficients in the expansion of a product of two Schur functions and carries an L-log-concavity property via Murota's discrete convex analysis.

If this is right

  • The Lam-Postnikov-Pylyavskyy conjecture holds for all such partitions.
  • Differences of Schur products expand with nonnegative coefficients in the Schur basis whenever the given conditions hold.
  • L-log-concavity of skeps transfers to Schur nonnegativity.
  • The same conditions guarantee that one product is at least as large as the other in the Schur partial order.

Where Pith is reading between the lines

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

  • Similar convex models might resolve other open positivity questions for symmetric functions.
  • Skeps could be used to derive new inequalities or bounds on Littlewood-Richardson coefficients beyond this result.
  • The discrete-convexity technique may apply to other combinatorial counts arising in representation theory.

Load-bearing premise

The skeps correctly enumerate the Littlewood-Richardson coefficients so that their L-log-concavity implies Schur nonnegativity of the difference.

What would settle it

Four partitions satisfying the sum equality and the min-max difference bounds for which the Schur expansion of s_λ' s_μ' − s_λ s_μ contains a negative coefficient.

Figures

Figures reproduced from arXiv: 2601.05007 by David E Speyer.

Figure 2.1
Figure 2.1. Figure 2.1: The left hand side depicts the skep inequalities: In each green parallelogram, and in all translates thereof, the sum of the + vertices is more than the sum of the − vertices; we also impose this condition on the green line segment. The right hand side, with the pink parallelograms, depicts the hive inequalities in the same manner. Example 2.3. As our running example of a skep, we will take g =     … view at source ↗
Figure 3
Figure 3. Figure 3: , edges in direction (1 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The restriction of the octahedron recurrence to the walls of T4. The four corners of the figure are labeled by the vertices of the tetrahedron T4. Fold along the dashed line; map the left side linearly to the wall i = 0 and map the right side linearly to the wall j = 0. Each edge is labelled with ehleft endpt − ehright endpt. The red edges indicate S bottom hive , S bottom skep , S top skep and S top hiv… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The projections of S top , S1, S2 and S ′ 1 to the (i, j)-plane (for n = 4). The numbers indicate the value of the t-coordinate. S1 = n[−1 i=0 n−[ 1−i j=0 ij ∪ n[−1 i=1 n[−i j=1 ij S2 = n[−1 i=0 n−[ 1−i j=1 ij ∪ n[−1 i=1 n−[ 1−i j=0 ij ∪ n[−1 k=0 k(n−1−k) Finally, let s = 0 for n even and s = 1 for n odd. We put S ′ 1 = S1 \ 00 ∪ Hull((0, 0, 2 − 3s),(1, 0, 1 − s),(0, 1, 1 − s)). In other words, S ′ 1 del… view at source ↗
Figure 3
Figure 3. Figure 3: shows the projections of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The intersections W1 ∩ S1 and W2 ∩ S2 (in bold), and some characteristic rhombus inequalities (in green). In the figure, we have n = 5, W1 = {t − 1 = −i + j} and W2 = {|t + 3| = i + j}. . Lemma 3.26. Let eh : Tn → Z be a function obeying the octahedron recurrence. Suppose that, for every wavefront W, there is a section S(W) transverse to W such that eh satisfies rhombus inequalities along W ∩ S(W). Then … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: On the left, we show Π(0000, 0235)/Z14, from Example 4.18. On the right, we show the symmetric functions from Example 4.20 [PITH_FULL_IMAGE:figures/full_fig_p018_4_1.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: The parallelepiped Π(0000, 0358)/Z14, discussed in Example 7.8. We observe that x bc y + x bc y = x + y so, if (λ, µ) ∈ Sπ, then (λ bc µ, λ bc µ) ∈ Sπ as well. It is also easy to check that (λ, µ) ⊑ (λ bc µ, λ bc µ), and that (λ, µ) ⊏ (λ bc µ, λ bc µ) as long as either b or c lies in [min(δi), max(δi)]. Theorem 7.5. For any (λ, µ) ⊒ (λ ′′, µ′′) in Sπ, we can find b < c such that (λ, µ) ⊐ (λ bc µ, λ bc µ)… view at source ↗
read the original abstract

Let $\lambda$, $\mu$, $\lambda'$, $\mu'$ be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if $\lambda+\mu = \lambda' + \mu'$, and $\min(\lambda_i-\lambda_j, \mu_i-\mu_j) \leq \lambda'_i - \lambda'_j \leq \max(\lambda_i-\lambda_j, \mu_i-\mu_j)$ for all $1 \leq i<j \leq n$, then $s_{\lambda'} s_{\mu'} - s_{\lambda} s_{\mu}$ is Schur nonnegative. We prove this conjecture. Our proof is based on two key ideas. First, we introduce a new combinatorial model for Littlewood-Richardson coefficients which we name ``skeps", which are similar to but distinct from Knutson and Tao's hives. Second, we use tools from Murota's theory of L-convexity to prove an L-log-concavity theorem for skeps.

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

2 major / 2 minor

Summary. The manuscript proves the Lam-Postnikov-Pylyavskyy conjecture: if partitions satisfy λ + μ = λ' + μ' and the min-max condition min(λ_i − λ_j, μ_i − μ_j) ≤ λ'_i − λ'_j ≤ max(λ_i − λ_j, μ_i − μ_j) for all i < j, then s_λ' s_μ' − s_λ s_μ is Schur nonnegative. The proof introduces a new combinatorial model called skeps (distinct from but similar to hives) claimed to enumerate Littlewood-Richardson coefficients, then applies Murota's L-convexity theory to establish L-log-concavity on skeps, which transfers to the required Schur nonnegativity.

Significance. If correct, the result resolves a longstanding conjecture in algebraic combinatorics concerning positivity of Schur function differences. The introduction of skeps as a new model and the importation of discrete-convexity tools from Murota's framework constitute a substantive methodological advance that could apply to other LR-coefficient positivity questions. The argument is self-contained once the skep-LR equivalence and the precise invocation of Murota's theorems are granted.

major comments (2)
  1. [Section introducing skeps] The central transfer step rests on skeps correctly enumerating Littlewood-Richardson coefficients. The manuscript must supply an explicit bijection or generating-function identity establishing this equivalence (presumably in the section introducing skeps); without it, L-log-concavity on skeps does not imply the claimed Schur nonnegativity.
  2. [Section on L-log-concavity for skeps] Application of Murota's L-convexity theorems: the manuscript invokes L-log-concavity results but must verify that the skep structure (with the given partial order or valuation) satisfies the exact hypotheses of the cited theorems from Murota (e.g., the required submodularity or L-convexity inequality). A direct check or reference to the precise statement used is needed to confirm the application is load-bearing.
minor comments (2)
  1. [Introduction] Clarify the precise distinction between skeps and Knutson-Tao hives in the introductory section to avoid potential confusion for readers familiar with hive models.
  2. Ensure all references to Murota's theorems include the exact theorem numbers or statements from the cited source for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our proof of the Lam-Postnikov-Pylyavskyy conjecture. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Section introducing skeps] The central transfer step rests on skeps correctly enumerating Littlewood-Richardson coefficients. The manuscript must supply an explicit bijection or generating-function identity establishing this equivalence (presumably in the section introducing skeps); without it, L-log-concavity on skeps does not imply the claimed Schur nonnegativity.

    Authors: We agree that the equivalence is foundational. The section introducing skeps already contains a generating-function identity equating the enumeration of skeps (with fixed boundary data) to the Littlewood-Richardson coefficient. To make this fully explicit as requested, we will add a dedicated subsection providing a direct bijection between skeps and LR tableaux, thereby rendering the transfer to Schur nonnegativity transparent. revision: yes

  2. Referee: [Section on L-log-concavity for skeps] Application of Murota's L-convexity theorems: the manuscript invokes L-log-concavity results but must verify that the skep structure (with the given partial order or valuation) satisfies the exact hypotheses of the cited theorems from Murota (e.g., the required submodularity or L-convexity inequality). A direct check or reference to the precise statement used is needed to confirm the application is load-bearing.

    Authors: We appreciate this request for verification. The L-log-concavity on skeps follows from Murota's theorem on L-convex functions once the valuation induced by the partial order is shown to be submodular. We will insert a direct check confirming that the skep structure satisfies the precise L-convexity inequality (referencing the exact statement, e.g., Theorem 3.2 in Murota's monograph) and thereby confirm that the hypotheses hold. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new model and external theory

full rationale

The paper introduces skeps as an independent combinatorial model for Littlewood-Richardson coefficients and establishes their correctness through direct enumeration arguments. It then applies Murota's pre-existing L-convexity framework (external to the paper) to derive L-log-concavity on these objects, which transfers to the target Schur nonnegativity under the stated partition conditions. No step reduces the conjecture to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain; the central claim rests on verifiable combinatorial equivalence and an independent discrete-convexity theorem rather than tautological input-output equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on standard facts about Schur functions and Littlewood-Richardson coefficients together with the new skep model and Murota's L-convexity results; no free parameters are introduced.

axioms (2)
  • standard math Schur functions form a basis for symmetric polynomials and their products expand with nonnegative Littlewood-Richardson coefficients
    Invoked to interpret the target difference as a linear combination whose coefficients must be shown nonnegative.
  • domain assumption Murota's theory of L-convexity implies L-log-concavity for certain discrete objects
    Applied directly to the skep model to obtain the key inequality.
invented entities (1)
  • skeps no independent evidence
    purpose: Combinatorial model for Littlewood-Richardson coefficients in the product of two Schur functions
    New object introduced to replace or augment hives; L-log-concavity is proved for skeps.

pith-pipeline@v0.9.0 · 5483 in / 1484 out tokens · 43059 ms · 2026-05-16T16:01:22.680453+00:00 · methodology

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Reference graph

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