Foulkes' conjecture for sixth powers and quadratic stabilization

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

Manuscripts (2)

Foulkes' conjecture for the sixth symmetric power

September 25, 2026 · source · cite as \cite{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026}

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 \mathop{\mathrm{GL}}\nolimits (V)-equivariant injection \mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V).

Quadratic stabilization of the canonical Foulkes--Howe map

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026}

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

Result 210 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.