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Math-2523 · mathematics · Algebra

Saturation factor one for even Spin groups

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $G=Spin(2n)$ over C, $n\geq 2$, and dominant integral weights $\lambda ,\mu ,\nu $ whose sum belongs to the root lattice, does the existence of a G-invariant in $V(N\lambda )\otimes V(N\mu )\otimes V(N\nu )$ for some integer $N\geq 1$ imply the existence of one in $V(\lambda )\otimes V(\mu )\otimes V(\nu )$?

Short English statement

For $G=Spin(2n)$ over C, $n\geq 2$, and dominant integral weights $\lambda ,\mu ,\nu $ whose sum belongs to the root lattice, does the existence of a G-invariant in $V(N\lambda )\otimes V(N\mu )\otimes V(N\nu )$ for some integer $N\geq 1$ imply the existence of one in $V(\lambda )\otimes V(\mu )\otimes V(\nu )$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2523-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2523

The main type-D saturation theorem proves equivalence of invariant existence before and after every positive dilation for dominant integral weights with sum in Q(D_n). The requested implication for some N is one direction of exactly that equivalence. Within these assumptions, this is a claimed affirmative answer to the stated question.

Saturation factor one for even Spin groups

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For $\operatorname{Spin}(2n)$, $n\geq2$, a tensor invariant at a dilated triple is claimed to imply one at the original triple when the weight sum belongs to the root lattice.

Mathematical statement of the claimed result

Result 1.
For $G=\operatorname{Spin}(2n)$, $n\ge2$, let $P_+$ be its dominant integral weights and $Q$ its type-$D_n$ root lattice. Take $\lambda,\mu,\nu\in P_+$ with $\lambda+\mu+\nu\in Q$. The claim is saturation with factor one: \[ (V(\lambda)\otimes V(\mu)\otimes V(\nu))^G\ne0 \quad\Longleftrightarrow\quad (V(N\lambda)\otimes V(N\mu)\otimes V(N\nu))^G\ne0 \] for every positive integer $N$. Here $V(\lambda)$ denotes the irreducible highest-weight module; the root-lattice condition is retained.

Outline of the argument

The proof constructs reflected weight schedules and resolves the type-D parity obstruction by joining signed-root components. Rank-two reorderings keep the modified schedule admissible, exterior and minuscule column constructions turn it into nonzero intertwiners, and a preprojective-module reduction removes common zero coordinates.

Formal verification

Family 204 has no Lean scope page in this release. Its selected result is not presented as formalized. 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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