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The qq-Foulkes problem II: Schur positivity with high probability

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

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Bergeron's qq-analog of Foulkes' conjecture asserts that (Hb[Ha]−Ha[Hb])/(1−q)(H_b[H_a]-H_a[H_b])/(1-q) is Schur positive for a≤ba\le b, where HnH_n is the (modified) Hall--Littlewood polynomial. We view this conjecture probabilistically: on each irreducible representation of SabS_{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 Cexp⁡(−c(ab)1/3)C\exp(-c(ab)^{1/3}), uniformly in a≤ba\le b. Consequently, the qq-Foulkes conjecture holds after deleting a fraction at most Cexp⁡(−c(ab)1/3)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.

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.

Tools used

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026