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Saxl’s conjecture

6 years old · traced to

Let ρ=(k,k−1,k−2,…,2,1)⊢n\rho=(k,k-1,k-2,\ldots,2,1)\vdash n be a staircase partition. Then

DC(ρ)⊗DC(ρ)=⨁λ⊢ng(ρ,ρ,λ)DC(λ),D_{\mathbb C}(\rho)\otimes D_{\mathbb C}(\rho)=\bigoplus_{\lambda\vdash n} g(\rho,\rho,\lambda)D_{\mathbb C}(\lambda),

with multiplicity g(ρ,ρ,λ)≠0g(\rho,\rho,\lambda)\ne 0 for all partitions λ\lambda of nn.

References

Source: 10.4171/OWR/2020/9, page 25. Canonicalized from 7 source-backed detection(s); richer provenance is retained in the corpus artifact.

Source: Oberwolfach Reports: 10.4171/OWR/2020/9, page 25

Posted · edited

Progress summary

Refreshed
Open · progress reported

A proposed proof was withdrawn after experts found gaps, while a separate conditional strengthening remains unverified, so the conjecture is open.

Saxl proposed in 2012 that the tensor square of the staircase representation contains every irreducible representation of the relevant symmetric group.

Known results and December 2025 withdrawal

  • Pak, Panova, and Vallejo (2013) proved occurrence for hooks and two-row partitions.
  • Ikenmeyer (2014) proved occurrence for every partition comparable with the staircase in dominance order, including all hooks.
  • Soong Kyum Lee’s December 2025 paper claimed a complete proof, but was withdrawn after expert reviewers identified mathematical gaps and stated that the claim was unjustified in its current form.
  • Later work gives further families and structural positivity results, but not positivity for every partition.

Community submission (unverified)

Posted October 25, 2026. A submitted manuscript argues conditionally that, for staircase size N=m(m+1)/2N=m(m+1)/2, one has g(ρ,ρ,λ)≥m−1g(\rho,\rho,\lambda)\ge m-1 except for the row and column partitions, with equality at (N−1,1)(N-1,1). It explicitly does not claim an independently established proof of Saxl’s conjecture.

Current status (as of October 2026): Saxl’s conjecture remains open; the 2025 claimed proof was withdrawn, and the submitted quantitative strengthening is unverified.

Sources

Solutions 1

Conditional multiplicity strengthening of Saxl's conjecture: for staircase ρ=(m,…,1), every nontrivial, nonsign constituent has Kronecker multiplicity at least m−1, with equality at (N−1,1). It relies on stronger stipulated source inputs, not just positivity.See full solutionHide full solution

Conditional strengthening: the sharp minimum multiplicity in a staircase tensor square.

The paper The sharp minimum multiplicity in a staircase tensor square studies ordinary complex tensor-square Kronecker coefficients. For m ≥ 2, N=m(m+1)/2 and ρ=(m,m−1,…,1), its conclusion is

g(ρ,ρ,λ)≥m−1g(\rho,\rho,\lambda)\ge m-1

for every partition λ of N other than the row (N) and column (1ᴺ). Equality holds for λ=(N−1,1), making the uniform floor sharp.

This is a conditional quantitative strengthening of the positivity question on this page. It assumes the manuscript's stipulated source cyclic-polytabloid, module-map, quotient and multiplicity-recursion inputs from OpenAI/math; it does not follow here from an independently established black-box positivity theorem. The additional deduction is the uniform quantitative bound through the four-exception invariant, predecessor bounds and low-capacity Kostka estimate.

This entry does not claim an independent proof of the original Saxl conjecture, nor a symmetric-square, exterior-square, integral or modular analogue. The Lean support concerns selected finite-poset and ordered-cut lemmas; the representation theory and full multiplicity theorem remain outside that formalization.

The full manuscript and supporting package are available on Zenodo, DOI 10.5281/zenodo.23647109.

Source and review status. The upstream inputs are stipulated at OpenAI/math revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. This is an AI-assisted manuscript with internal project reviews, not external peer review. The supporting Lean checks cover selected lemmas, not the full theorem.

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MJ Kang Posted