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The qq-Foulkes problem I: A fusion inequality for Macdonald plethysms

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RT — Representation Theory

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove that the plethysm Hm[H~μ]H_m[\widetilde H_\mu] of a Macdonald polynomial by the Hall–Littlewood polynomial HmH_m lies above the Macdonald polynomial H~mμ\widetilde H_{m\mu} obtained by fusing mm copies of H~μ\widetilde H_\mu. More specifically, the quotient (Hm[H~μ]−H~mμ)/(1−qm)(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m) is Schur positive with coefficients in N[q,t]\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 qq-Foulkes plethysms Hb[Ha]H_b[H_a] and Ha[Hb]H_a[H_b] above the same polynomial HabH_{ab}, and comparing their two defects we obtain a distorted version of Bergeron's qq-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by qa(a−1)⌊b/2⌋q^{a(a-1)\lfloor b/2\rfloor}.

Provenance statement

The results in this paper were developed through extensive use of AI assistants in sustained mathematical discussion, including failed attempts, counterexamples, audits, and repairs. AI assistants also helped with exact computations, proof checks, and exposition. The external representation-theoretic results are cited at their points of use. The main text has not yet been vetted line by line by the author.

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026