Foulkes' conjecture for sixth powers and quadratic stabilization

What OpenAI claims

Proves the sixth case of Foulkes’ conjecture: Sym ⁶(Sym ^bV) embeds equivariantly in Sym ^b(Sym ⁶V) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map Sym ^b(Sym ^aV) → Sym ^a(Sym ^bV) is surjective for a ≥ 2 and b ≥ a(a-1), giving dimension-independent quadratic stabilization. (Lean)

Papers (2)

1. Foulkes' conjecture for the sixth symmetric power

   25 September 2026 · Preprint, not peer reviewed · PDF

   We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a GL (V)-equivariant injection Sym ⁶(Sym ^b V) ↪ Sym ^b(Sym ⁶ V).

2. Quadratic stabilization of the canonical Foulkes--Howe map

   25 September 2026 · Preprint, not peer reviewed · PDF

   For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map Sym ^b(Sym ^a V) ⟶ Sym ^a(Sym ^b V) is surjective whenever a ≥ 2 and b ≥ a(a-1). This gives a quadratic stabilization bound independent of dim V.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.