Quadratic stabilization of the Foulkes–Howe map
OpenAI's theorem (2026) that the canonical Hermite–Hadamard–Howe map Sym^b(Sym^a V) -> Sym^a(Sym^b V) is surjective for b >= a(a-1) and every finite-dimensional complex V. Symmetric powers and averaging, the image of the map, the diagonal criterion for surjectivity, an injectivity lemma for directional derivatives with the Fischer inner product, the induction that proves the vanishing on products, and an equivariant embedding of the kind asked for by Foulkes' conjecture.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Linear algebra over the complex numbers: tensor products, quotient spaces, dual spaces, Hermitian inner products and adjoints.
- Polynomials in several variables and their partial derivatives.