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On majorization inequalities for normalized Jack and Macdonald polynomials

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.CA — Classical Analysis and ODEsmath.RT — Representation Theory

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We study the inequality Pλ(x;q,t)/Pλ(1n;q,t)≥Pμ(x;q,t)/Pμ(1n;q,t)P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t) for all x∈R>0nx\in\mathbb{R}_{>0}^n whenever the partition λ\lambda majorizes μ\mu; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every nn from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case n=2n=2, the covers (d)⪰(d−1,1)(d)\succeq(d-1,1) for every nn when t≤qt\leq q, and, by exact certificates, the case n=3n=3 with ∣λ∣≤8|\lambda|\leq 8. The HCIZ mechanism behind the Jack proof has no direct finite-qq analogue: at two explicit pairs (q,t)(q,t), no positive representation of the normalized ratio has a kernel multiplicative in λ\lambda, and normalizing at the principal specialization tδt^\delta instead fails the inequality.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs.

Tools used

OpenAI
GPT Pro

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026