The tensor square conjecture for symmetric groups: some irreducible representation of S_n has a tensor square containing every irreducible

For partitions of $n$ let $g(\lambda,\mu,\nu)$ be the Kronecker coefficients of $S_n$ . Heide, Saxl, Tiep and Zalesski showed that for most finite simple groups of Lie type the Steinberg character has a tensor square containing every irreducible character. For symmetric groups, Pak, Panova and Vallejo formulated the tensor square conjecture (Conjecture 1.1): for every $n\ge3$ other than $4$ and $9$ there is $\lambda\vdash n$ with $g(\lambda,\lambda,\nu)>0$ for all $\nu\vdash n$ ; it fails for $n=2,4,9$ . Saxl's conjecture would give it at triangular $n$ ; elsewhere only partial and asymptotic results were known (Luo and Sellke found irreducibles whose fourth tensor power contains everything). Does every $S_n$ with $n\notin\{2,4,9\}$ have an irreducible representation whose tensor square contains every irreducible representation?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Representation theory of symmetric groups; Kronecker coefficients
Posed by: Igor Pak, Greta Panova and Ernesto Vallejo
Year posed: 2013
Years open: 13y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 28 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every positive integer $n\notin\{2,4,9\}$ there is $\lambda\vdash n$ , necessarily self-conjugate, with $g(\lambda,\lambda,\nu)>0$ for every $\nu\vdash n$ . Corollary 1.2 transfers this through Letellier's results to unipotent characters of $GL_n(\mathbb F_q)$ for every prime power $q$ . The construction combines the cyclic Saxl theorem with band, balance and capacity arguments, and a finite verification for small degrees. Not shown: an explicit closed-form family of such $\lambda$ at nontriangular degrees beyond the construction, or bounds on the coefficients.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'Universal Tensor Squares for Symmetric Groups' (September 24, 2026). It uses the cyclic form of Saxl's conjecture from the companion manuscript and a finite computer verification (C++ and Python) for small degrees, shipped in the release.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Universal Tensor Squares for Symmetric Groups' was read against Pak-Panova-Vallejo's Conjecture 1.1; it is the conjecture, including nontriangular degrees. Lean: lean/ComparatorChallenges/UniversalTensorSquares.json exists with solution module OAI.RepresentationTheory.UniversalSquare.Main present at the pinned commit; it is not in the formalization catalogue (formalization.yaml). The statement read here asserts, for every $n>0$ with $n\ne2,4,9$ , a Young diagram $\lambda$ of size $n$ with positive Kronecker multiplicity against every $\nu$ , irreducibility of its Specht representation, and an injective intertwiner from every finite-dimensional irreducible complex representation of $S_n$ into its tensor square. That states the headline. Not rebuilt here, and the release's finite character computations were not run.

The tensor square conjecture for symmetric groups: some irreducible representation of S_n has a tensor square containing every irreducible

What was actually shown

What the AI did

Verification

Sources

Discussion