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The $q$-Foulkes problem II: Schur positivity with high probability
Abstract
Bergeron's $q$-analog of Foulkes' conjecture asserts that $(H_b[H_a]-H_a[H_b])/(1-q)$ is Schur positive for $a\le b$, where $H_n$ is the (modified) Hall–Littlewood polynomial. We view this conjecture probabilistically: on each irreducible representation of $S_{ab}$, the two plethysms induce degree distributions, and positivity means that the first is stochastically smaller than the second. We prove that, for a random (Plancherel) irreducible representation and a degree sampled from its graded multiplicity, the corresponding coefficient is negative with probability at most $C\exp(-c(ab)^{1/3})$, uniformly in $a\le b$. Consequently, the $q$-Foulkes conjecture holds after deleting a fraction at most $C\exp(-c(ab)^{1/3})$ of each of the two graded modules. Additionally, we show that exact conditional means and variances yield complete positivity for explicit width and height families.
Versions
Version numbers are this site's own; arXiv and Hexagon numbering, where present, are separate. Every released file carries an OpenTimestamps proof made on its release day.
| Version | Released | Length | File | Proof | arXiv | Hexagon |
|---|---|---|---|---|---|---|
| v1 | 28 October 2026 | 32 pp | .ots | — | — |
Cite
@misc{sra2026qfoulkes2-v1,
author = {Suvrit Sra},
title = {{The $q$-Foulkes problem II: Schur positivity with high probability}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qfoulkes2/v1}},
note = {Version v1, released 28 October 2026},
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Verify the timestamp
pip install opentimestamps-client curl -O https://suvrit.github.io/papers/qfoulkes2/qfoulkes2-v1.pdf -O https://suvrit.github.io/papers/qfoulkes2/qfoulkes2-v1.pdf.ots ots verify qfoulkes2-v1.pdf.ots