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Saxl's conjecture: the tensor square of the staircase representation contains every irreducible representation of the symmetric group

For partitions λ,μ,ν\lambda,\mu,\nu of nn the Kronecker coefficient g(λ,μ,ν)=dim⁡HomSn(Sν,Sλ⊗Sμ)g(\lambda,\mu,\nu)=\dim\mathrm{Hom}_{S_n}(S^\nu,S^\lambda\otimes S^\mu) is the multiplicity of the Specht module SνS^\nu in Sλ⊗SμS^\lambda\otimes S^\mu. Let ρm=(m,m−1,…,1)\rho_m=(m,m-1,\ldots,1) be the staircase partition of Nm=m(m+1)/2N_m=m(m+1)/2. At the UCLA Combinatorics Seminar on 20 March 2012 Saxl proposed that Sρm⊗SρmS^{\rho_m}\otimes S^{\rho_m} contains every irreducible representation of SNmS_{N_m} (recorded by Pak, Panova and Vallejo, Conjecture 1.2). Known cases covered hooks, two-row shapes, double and triple hooks, partitions comparable to ρm\rho_m in dominance order, and random partitions with high probability. Is g(ρm,ρm,λ)>0g(\rho_m,\rho_m,\lambda)>0 for every m≥1m\ge1 and every partition λ⊢Nm\lambda\vdash N_m?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Representation theory of symmetric groups; Kronecker coefficients
Posed by
Jan Saxl
Year posed
2012
Years open
14y
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
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every m≥1m\ge1 and every λ⊢Nm=m(m+1)/2\lambda\vdash N_m=m(m+1)/2, g(ρm,ρm,λ)>0g(\rho_m,\rho_m,\lambda)>0, so the tensor square of the staircase Specht module contains every irreducible representation of SNmS_{N_m}. The proof gives the stronger statement that every constituent occurs inside one explicitly generated cyclic submodule of the tensor square, extending Ikenmeyer's triangular arrangement of alternating tensors. Not shown: values or bounds for these Kronecker coefficients, or a combinatorial interpretation of them.

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 'A Cyclic Polytabloid Proof of Saxl's Conjecture' (September 24, 2026); the companion 'Universal Tensor Squares for Symmetric Groups' of the same date uses its cyclic statement and is entered separately.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture as Pak, Panova and Vallejo record it; it is the conjecture verbatim. Lean: formalization.yaml lists comparator ComparatorChallenges/Saxl.json, declaration OAI.Saxl.saxl_conjecture in OAI/RepresentationTheory/Saxl/Main.lean. The statement file was read here: Specht modules are orbit spans of polytabloids in a complex word space, the Kronecker multiplicity is the dimension of the space of intertwining maps from SμS^\mu into Sρm⊗SρmS^{\rho_m}\otimes S^{\rho_m}, and SaxlConjecture asserts positivity for every m≥1m\ge1 and every Young diagram of size m(m+1)/2m(m+1)/2. That states the headline. Not rebuilt here. The paper's stronger cyclic-submodule statement is not formalised.

Sources

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