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Math-2511 · mathematics · Algebra

A dimension-free quadratic bound for the canonical Foulkes–Howe map

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For all integers $a\geq 2$ and $b\geq a(a-1)$, and all finite-dimensional complex V, is the canonical multiplication-and-averaging map $Sym^{b}(Sym^{a}V)\to Sym^{a}(Sym^{b}V)$ surjective?

Short English statement

For all integers $a\geq 2$ and $b\geq a(a-1)$, and all finite-dimensional complex V, is the canonical multiplication-and-averaging map $Sym^{b}(Sym^{a}V)\to Sym^{a}(Sym^{b}V)$ surjective?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2511-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2511

The paper defines mu_{a,b,V} by multiplication after the averaging identifications, exactly the stated canonical map, and proves surjectivity for all a>=2, b>=a(a-1) and finite-dimensional complex V. lean/docs/210.md documents this exact map/range. Within these assumptions, this is a claimed affirmative answer to the stated question.

A dimension-free quadratic bound for the canonical Foulkes–Howe map

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The canonical Foulkes--Howe map is claimed onto whenever $a\geq2$ and $b\geq a(a-1)$, uniformly in the underlying dimension.

Mathematical statement of the claimed result

Result 1.
Let $V$ be any finite-dimensional complex vector space. Using averaging over permutations to identify symmetric powers with invariant tensors, multiplication in $(\operatorname{Sym}V)^{\otimes a}$ defines the canonical Foulkes--Howe map \[ \mu_{a,b,V}:\operatorname{Sym}^b(\operatorname{Sym}^aV) \longrightarrow\operatorname{Sym}^a(\operatorname{Sym}^bV). \] The asserted dimension-independent sufficient bound is \[ a\ge2,\quad b\ge a(a-1)\quad\Longrightarrow\quad\mu_{a,b,V}\text{ surjective}. \]

Outline of the argument

An annihilator formulation reduces surjectivity to vanishing of symmetric multilinear forms. Two successive differential procedures first free a slot from common factors and then eliminate the remaining common power; injectivity of directional differentiation supplies the quadratic degree condition.

Formal verification

For family 210, the formal scope is canonical surjectivity in the quadratic range, its $a=1$ case, and reverse embeddings. For sixth powers this only covers $b\geq30$, rather than every $b\geq6$. The listed Comparator files are FoulkesHowe.lean. Named declarations include OAI.Problem346.canonical_foulkes_howe_surjective. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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