Results on the Foulkes–Howe map: averaging, the diagonal criterion, the injectivity lemma and the vanishing theorem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Quadratic stabilization (OpenAI) — Theorem A. For a >= 2, b >= a(a-1) and every finite-dimensional complex V, the map Sym^b(Sym^a V) -> Sym^a(Sym^b V) is surjective. Proved in the last lesson.
• An equivariant embedding (OpenAI) — Corollary B. For a >= 2 and b >= a(a-1) there is an injective GL(V)-equivariant linear map Sym^a(Sym^b V) -> Sym^b(Sym^a V). Proved in the last lesson.
• Averaging — Lemma 1.1. Averaging identifies Sym^k W with the symmetric tensors in W^(x)k, naturally in W.
• Polarization — Lemma 1.2. k! u_1...u_k is the alternating sum of (sum over J of u_j)^k over subsets J; pure powers span Sym^k W.
• The image — Proposition 2.2. The image of the Foulkes–Howe map is spanned by the powers P^a of products P of b vectors.
• The diagonal criterion (OpenAI) — Proposition 3.1. The map is surjective if and only if every symmetric a-linear form on Sym^b V vanishing at (P, ..., P) for all products P of b vectors is zero.
• Fischer inner product — Lemma 2.1. Differentiation by X_k and multiplication by X_k are adjoint for the inner product with ||X^alpha||^2 = alpha!.
• Differential injectivity (OpenAI) — Lemma 3.1. On polynomials homogeneous of degree p in U and q in Y, possibly in further variables, the operator sum_k Y_k d/dU_k is injective when p > q.
• Vanishing on products (OpenAI) — Proposition 1.1. For b >= a(a-1), a symmetric a-linear form on Sym^b V vanishing on the diagonal at all products of b vectors is zero.

Reading order

Read the lessons in order. The second lesson is independent of the first and can be read on its own; the third uses both.