we-solve (beta)

Math-2524 · mathematics · Algebra

An irreducible tensor square containing every symmetric-group representation

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each positive integer n outside{2,4,9}, does $S_{n}$ have an irreducible complex representation V such that $V\otimes V$ contains every irreducible complex representation of $S_{n}$?

Short English statement

For each positive integer n outside{2,4,9}, does $S_{n}$ have an irreducible complex representation V such that $V\otimes V$ contains every irreducible complex representation of $S_{n}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2524-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2524

The universal tensor-square theorem supplies a partition lambda for each positive n outside {2,4,9}, with g(lambda,lambda,nu)>0 for every nu. lean/docs/205.md separately documents this exact universal result. Within these assumptions, this is a claimed affirmative answer to the stated question.

An irreducible tensor square containing every symmetric-group representation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For each symmetric-group degree other than $2,4,9$, the authors claim an irreducible representation whose square contains every irreducible constituent.

Mathematical statement of the claimed result

Result 1.
For every positive integer $n\notin\{2,4,9\}$, the source claims a partition $\lambda\vdash n$ satisfying \[ \forall\nu\vdash n,\qquad g(\lambda,\lambda,\nu)>0. \] Thus one irreducible complex representation of $S_n$ has every irreducible as a constituent of its tensor square. The three exceptional sizes are excluded from this theorem.

Outline of the argument

The construction selects a self-conjugate partition in each permitted degree and extends the cyclic polytabloid method beyond staircases. Combinatorial decomposition and exact capacity checks cover the remaining degrees; positivity transfers to corresponding unipotent tensor-square multiplicities by the cited character-theoretic comparison.

Formal verification

For family 205, the Lean scope includes every staircase tensor square and the universal irreducible squares away from degrees $2,4,9$; occurrence inside the orbit of the prescribed cyclic vector is extra manuscript scope. The listed Comparator files are Saxl.lean, UniversalTensorSquares.lean. Named declarations include OAI.Saxl.saxl_conjecture. 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

Go to contribution →