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The qq-Foulkes problem III: On Foulkes' conjecture with a graded inner factor

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

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We show that surjectivity of the canonical map Symb(SymaV)→Syma(SymbV)\mathrm{Sym}^b(\mathrm{Sym}^aV)\to\mathrm{Sym}^a(\mathrm{Sym}^bV) survives grading of the inner factor. Consequently hb[Ha]−Ha[hb]h_b[H_a]-H_a[h_b] is Schur positive whenever the canonical map is surjective, where HaH_a is the (modified) Hall--Littlewood polynomial; the same holds for (modified) Macdonald polynomials. We prove this surjectivity for b≥(a−1)2b\ge(a-1)^2, improving the known bound a(a−1)a(a-1). For Bergeron's qq-analog of Foulkes' conjecture that lets both factors be graded, we prove nonnegativity of every Schur coefficient for certain special cases. We also study the full conjecture under an expansion hypothesis on an explicit space of maps.

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. v1 · 18 pagesSubmitted by Suvrit SraInitial depositCurrentOct 31, 2026