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Foulkes' conjecture for the sixth symmetric power: Sym6(SymbV)↪Symb(Sym6V)\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V)

For a finite-dimensional complex vector space VV and integers 1≤a≤b1\le a\le b, Foulkes' conjecture (1950) asks for a GL(V)\mathrm{GL}(V)-equivariant injection Syma(SymbV)↪Symb(SymaV)\mathrm{Sym}^a(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^aV), equivalently that the plethysm difference hb[ha]−ha[hb]h_b[h_a]-h_a[h_b] is Schur-positive. It was known for a≤5a\le5: a=2a=2 from Thrall, a=3a=3 by Dent-Siemons, a=4a=4 and a=5a=5 from McKay's propagation theorem with computations by Müller-Neunhöffer and Cheung-Ikenmeyer-Mkrtchyan. Brion proved it for bb large in terms of aa, and Evseev-Paget-Wildon checked a+b≤19a+b\le19. The canonical Foulkes-Howe map has a kernel at (6,6)(6,6), so the propagation route fails at a=6a=6. For a=6a=6, is there an equivariant injection Sym6(SymbV)↪Symb(Sym6V)\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V) for every b≥6b\ge6 and every VV?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Representation theory; plethysm
Posed by
H. O. Foulkes (J. London Math. Soc. 1950)
Year posed
1950
Years open
76y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every b≥6b\ge6 and every finite-dimensional complex VV there is a GL(V)\mathrm{GL}(V)-equivariant injection Sym6(SymbV)↪Symb(Sym6V)\mathrm{Sym}^6(\mathrm{Sym}^bV)\hookrightarrow\mathrm{Sym}^b(\mathrm{Sym}^6V), i.e. hb[h6]−h6[hb]h_b[h_6]-h_6[h_b] is Schur-positive. The range b≥30b\ge30 comes from the companion's quadratic stabilization theorem; 6≤b≤296\le b\le29 is settled by certified comparisons of upper bounds for source multiplicities with lower bounds for target multiplicities. It proves the multiplicity comparison, not injectivity of the canonical map (which fails at (6,6)(6,6)), and only the case a=6a=6; Foulkes' conjecture for a≥7a\ge7 and b<a(a−1)b<a(a-1) stays open.

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The proof combines the companion's quadratic-stabilization theorem (range b >= 30) with certified multiplicity comparisons computed by printed programs for 6 <= b <= 29; a second, longer route runs the certificates through b = 149 and uses a chart bound beyond.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the introduction were read against Foulkes' conjecture as the manuscript states it. Lean: lean/docs/210.md covers only the companion (challenge FoulkesHowe, theorem OAI.Problem346.canonical_foulkes_howe_surjective, in formalization.yaml). Its statement gives surjectivity of the canonical map for b≥a(a−1)b\ge a(a-1), so for a=6a=6 only b≥30b\ge30, and states surjectivity rather than the embedding; the docs say the companion's full range b≥6b\ge6 is not included. So the formal statement covers a special range, not this entry's headline, and the tier stays unreviewed. The band 6≤b≤296\le b\le29 rests on computer certificates that were not rerun here. Not refereed.

Sources

Changelog1 change

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