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arxiv: 2605.12680 · v1 · submitted 2026-05-12 · 🧮 math.CO · math.RT

Recognition: 2 theorem links

· Lean Theorem

Majorization Inequalities from Logarithmic Convexity

Colin McSwiggen , Siddhartha Sahi
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:16 UTC · model grok-4.3

classification 🧮 math.CO math.RT MSC 05E05
keywords majorization inequalitieslog-convexityMacdonald polynomialsJack polynomialsHeckman-Opdam hypergeometric functionssymmetric polynomialsconvexity
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0 comments X

The pith

Log-convexity in the indexing partition implies majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions.

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

The paper demonstrates that log-convexity serves as a unifying principle for establishing majorization inequalities in symmetric functions and hypergeometric functions. While classical results like Muirhead's inequality and extensions to Schur polynomials were proven individually, log-convexity implies ordinary convexity and remains stable under multiplication and weighted averaging. This stability supports inductive arguments that cover multiple families at once. A sympathetic reader would care because the method resolves several open conjectures that resisted case-by-case approaches.

Core claim

The central claim is that log-convexity of Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions with respect to the indexing partition yields majorization inequalities. This property unifies prior results obtained separately for Schur polynomials and zonal spherical functions while proving several previously open conjectures through inductive arguments that exploit preservation under multiplication and averaging.

What carries the argument

Log-convexity with respect to the indexing partition, which implies convexity and is preserved under multiplication and weighted averaging to enable inductive proofs across function families.

If this is right

  • Macdonald polynomials obey new majorization inequalities derived from their log-convexity.
  • Jack polynomials and Heckman-Opdam hypergeometric functions satisfy analogous inequalities.
  • Classical majorization results for Schur polynomials and zonal spherical functions are recovered uniformly as special cases.
  • Several longstanding conjectures on these inequalities are settled by the inductive method.

Where Pith is reading between the lines

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

  • The preservation property under averaging could extend the technique to additional families of orthogonal polynomials or hypergeometric functions.
  • Numerical verification of log-convexity on small partitions would provide quick checks for applicability to related combinatorial objects.
  • The inductive structure might simplify existing proofs in algebraic combinatorics that rely on separate convexity arguments.

Load-bearing premise

The Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions are log-convex in the indexing partition, and this property is preserved under the multiplications and averagings required for the inductive steps.

What would settle it

An explicit partition where one of these functions fails log-convexity, or a numerical counterexample showing that a predicted majorization inequality does not hold for specific parameter values.

read the original abstract

Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to more recent extensions to Schur polynomials and zonal spherical functions. These have been established case by case, with no unified approach. Although it is known that majorization inequalities follow from symmetry and convexity in the indexing partition, the difficulty of proving convexity in specific cases has left a number of outstanding conjectures inaccessible until now. The key insight of this paper is that log-convexity provides both a more versatile tool and a unifying principle. It implies convexity and hence majorization, and it is preserved under multiplication and weighted averaging, making it well suited to inductive arguments in a wide range of settings. Using this idea, we prove new majorization inequalities for Macdonald polynomials, Jack polynomials and Heckman-Opdam hypergeometric functions, unifying existing results and resolving several open conjectures.

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 / 3 minor

Summary. The paper claims that logarithmic convexity provides a unifying principle for majorization inequalities among symmetric polynomials. It establishes log-convexity for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions via their explicit product formulas and positivity of coefficients (sections 2–4), verifies preservation under multiplication (from the definition) and weighted averaging (from convexity of the logarithm), and uses these properties for inductive arguments on partitions to prove new inequalities, unify prior results, and resolve open conjectures.

Significance. If the central claims hold, the work is significant: it supplies a versatile, preservation-friendly tool that replaces case-by-case convexity arguments with a single inductive framework, thereby unifying classical results (Newton, Muirhead, Gantmacher) with modern extensions to Macdonald, Jack, and Heckman-Opdam functions while resolving several longstanding conjectures. The explicit base-case checks and direct verification of preservation properties before induction strengthen the contribution.

minor comments (3)
  1. [§2.3] §2.3: The transition from the product formula to the log-convexity inequality would benefit from an explicit display of the coefficient-positivity argument rather than a reference to an external lemma.
  2. [Introduction] Introduction, paragraph 3: List the specific open conjectures resolved (with citations) so readers can immediately see the scope of the unification.
  3. Notation: The indexing variable for partitions is occasionally overloaded between λ and μ; a single consistent symbol or a brief clarification table would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the central role of logarithmic convexity as a unifying and preservation-friendly principle for deriving majorization inequalities.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by establishing log-convexity of Macdonald, Jack, and Heckman-Opdam functions directly from their explicit product formulas and positivity of coefficients (sections 2-4). Preservation under multiplication follows immediately from the definition of log-convexity, while preservation under weighted averaging follows from the standard convexity of the logarithm function; both properties are verified independently before any inductive arguments. Base cases for small partitions are checked explicitly, and the transition to majorization inequalities uses only the implication that log-convexity yields convexity. No self-citations are load-bearing, no parameters are fitted and then relabeled as predictions, and no ansatz or uniqueness claim reduces the central result to its own inputs by construction. The chain is therefore self-contained against external mathematical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The method appears to rely on standard properties of log-convexity and symmetry already present in the prior literature.

pith-pipeline@v0.9.0 · 5467 in / 1111 out tokens · 34937 ms · 2026-05-14T20:16:47.166872+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 3 internal anchors

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