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Math-2515 · mathematics · Algebra

Foulkes inclusion for the sixth symmetric power

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every finite-dimensional complex vector space V and every integer $b\geq 6$, does there exist a GL(V)-equivariant injection $Sym^{6}(Sym^{b}V)\to Sym^{b}(Sym^{6}V)$?

Short English statement

For every finite-dimensional complex vector space V and every integer $b\geq 6$, does there exist a GL(V)-equivariant injection $Sym^{6}(Sym^{b}V)\to Sym^{b}(Sym^{6}V)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2515-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2515

The primary theorem gives the requested equivariant injection for every b>=6 and every finite-dimensional complex V, including 6<=b<30. lean/docs/210.md explicitly restricts the formal sixth-power range to b>=30 and excludes this full companion range. Within these assumptions, this is a claimed affirmative answer to the stated question.

Foulkes inclusion for the sixth symmetric power

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The sixth-power Foulkes comparison is claimed for every second exponent $b\geq6$ and every finite complex dimension.

Mathematical statement of the claimed result

Result 1.
For each integer $b\ge6$ and finite-dimensional complex $V$, the source claims a $\operatorname{GL}(V)$-equivariant inclusion \[ \operatorname{Sym}^6(\operatorname{Sym}^bV) \hookrightarrow\operatorname{Sym}^b(\operatorname{Sym}^6V). \] Equivalently, $h_b[h_6]-h_6[h_b]$ has nonnegative Schur coefficients, where brackets denote plethysm and $h_j$ the complete symmetric function.

Outline of the argument

Large degrees follow from canonical-map stabilization. For the remaining degrees the proof compares source character bounds with independently amplified target multiplicities, uses highest-weight shifts to reduce partitions, and finishes a narrow residual band by exact rational-function certificates.

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 $b\geq6$. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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.

Sources

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