Research Preprints · Combinatorics
The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms
Abstract
We prove that the plethysm $H_m[\widetilde H_\mu]$ of a Macdonald polynomial by the Hall–Littlewood polynomial $H_m$ lies above the Macdonald polynomial $\widetilde H_{m\mu}$ obtained by fusing $m$ copies of $\widetilde H_\mu$. More specifically, the quotient $(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m)$ is Schur positive with coefficients in $\mathbb N[q,t]$, and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both $q$-Foulkes plethysms $H_b[H_a]$ and $H_a[H_b]$ above the same polynomial $H_{ab}$, and comparing their two defects we obtain a distorted version of Bergeron's $q$-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by $q^{a(a-1)\lfloor b/2\rfloor}$.
Versions
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 | 26 October 2026 | 20 pp | .ots | — |
Cite
@misc{sra2026qfoulkes1-v1,
author = {Suvrit Sra},
title = {{The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qfoulkes1/v1}},
note = {Version v1, released 26 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/qfoulkes1/qfoulkes1-v1.pdf -O https://suvrit.github.io/papers/qfoulkes1/qfoulkes1-v1.pdf.ots ots verify qfoulkes1-v1.pdf.ots