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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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References

  1. Fran ç ois Bergeron. A qq -analog of F oulkes' conjecture. The Electronic Journal of Combinatorics , 24 0 (1): 0 Paper 1.38, 2017. 10.37236/6004.DOI
  2. Fran ç ois Bergeron, Adriano M. Garsia, Mark Haiman, and Glenn Tesler. Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods and Applications of Analysis , 6 0 (3): 0 363--420, 1999. 10.4310/MAA.1999.v6.n3.a7.DOI
  3. Michel Brion. Stable properties of plethysm: on two conjectures of F oulkes. Manuscripta Mathematica , 80: 0 347--371, 1993. 10.1007/BF03026558.DOI
  4. Man-Wai Cheung, Christian Ikenmeyer, and Sevak Mkrtchyan. Symmetrizing tableaux and the 5th case of the F oulkes conjecture. Journal of Symbolic Computation , 80: 0 833--843, 2017. 10.1016/j.jsc.2016.09.002.DOI
  5. Suzie C. Dent and Johannes Siemons. On a conjecture of F oulkes. Journal of Algebra , 226 0 (1): 0 236--249, 2000. 10.1006/jabr.1999.8169.DOI
  6. H. O. Foulkes. Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society , s1-25 0 (3): 0 205--209, 1950. 10.1112/jlms/s1-25.3.205.DOI
  7. Eugenio Giannelli. On the decomposition of the F oulkes module. Archiv der Mathematik , 100 0 (3): 0 201--214, 2013. 10.1007/s00013-013-0496-1.DOI
  8. Mark Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society , 14 0 (4): 0 941--1006, 2001. 10.1090/S0894-0347-01-00373-3.DOI
  9. Ian G. Macdonald. Symmetric Functions and Hall Polynomials . Oxford University Press, Oxford, 2nd edition, 1995.
  10. Tom McKay. On plethysm conjectures of S tanley and F oulkes. Journal of Algebra , 319 0 (5): 0 2050--2071, 2008. 10.1016/j.jalgebra.2007.12.003.DOI
Show all 15 referencesShow fewer references
  1. OpenAI . Foulkes ' conjecture for the sixth symmetric power, 2026 a . URL https://www.emergentmind.com/openai-math-explorer/papers/373.pdf. OpenAI Math Explorer, manuscript 373, 25 September 2026.link
  2. OpenAI . Quadratic stabilization of the canonical Foulkes--Howe map, 2026 b . Preprint, 25 September 2026. Manuscript (https://github.com/openai/math/blob/main/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf).link
  3. Suvrit Sra. The qq - F oulkes problem II : Schur positivity with high probability. Preprint, 2026 a .
  4. Suvrit Sra. Supplement to the qq - F oulkes project: Additional results and certificates. Preprint, 2026 b .
  5. R. M. Thrall. On symmetrized K ronecker powers and the structure of the free L ie ring. American Journal of Mathematics , 64: 0 371--388, 1942. 10.2307/2371691.DOI

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026