Research Preprints · Combinatorics
On majorization inequalities for normalized Jack and Macdonald polynomials
Abstract
We study the inequality $P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)$ for all $x\in\mathbb{R}_{>0}^n$ whenever the partition $\lambda$ majorizes $\mu$; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every $n$ from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case $n=2$, the covers $(d)\succeq(d-1,1)$ for every $n$ when $t\leq q$, and, by exact certificates, the case $n=3$ with $|\lambda|\leq 8$. The HCIZ mechanism behind the Jack proof has no direct finite-$q$ analogue: at two explicit pairs $(q,t)$, no positive representation of the normalized ratio has a kernel multiplicative in $\lambda$, and normalizing at the principal specialization $t^\delta$ instead fails the inequality.
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Version numbers are this site's own; arXiv numbering, where present, is separate. Every released file carries an OpenTimestamps proof made on its release day.
| Version | Released | Length | File | Proof | arXiv |
|---|---|---|---|---|---|
| v1 | 8 October 2026 | 20 pp | .ots | — |
Cite
@misc{sra2026jackratio-v1,
author = {Suvrit Sra},
title = {{On majorization inequalities for normalized Jack and Macdonald polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/jackratio/v1}},
note = {Version v1, released 8 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/jackratio/jackratio-v1.pdf -O https://suvrit.github.io/papers/jackratio/jackratio-v1.pdf.ots ots verify jackratio-v1.pdf.ots