Result 205, Algebra

Saxl’s conjecture and universal tensor squares

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S_n with $n\notin\{2,4,9\}$ has an irreducible representation whose tensor square contains all irreducibles.

Lean formalization Proof

The bigger picture

Why it matters

Representations describe permutations as linear transformations; irreducible ones are their basic building blocks. Two unreviewed manuscripts claim that, for broad families of symmetric groups, one such block combined with itself contains every possible block.

What changes?

The symmetric group Sn consists of all permutations of n objects. A tensor square combines a representation with itself. The staircase claim covers the irreducible complex representation indexed by rows of lengths k, k-1, down to 1, with n equal to their sum: its tensor square contains every irreducible of that group. The broader claim guarantees some irreducible complex representation with this property for every positive integer n other than 2, 4, and 9.

What does that help mathematicians do?

This turns separate containment questions, one for each irreducible, into a uniform guarantee. At staircase sizes, it also identifies the representation to use, rather than merely asserting that one exists. Researchers could therefore rule out any missing irreducible when decomposing that tensor square into basic pieces. The claim concerns presence, not equal multiplicities or exactly one copy of each component.

Are there practical applications?

The immediate value is foundational: the claims describe how the basic types of permutation symmetry arise together through tensor products. They would give researchers a single representation whose square encompasses all those types at each covered size. The supplied claims do not establish an efficient procedure for constructing or separating the resulting components.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Universal Tensor Squares for Symmetric Groups

September 24, 2026 37 pages

For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of S_n has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026,
  author = {{OpenAI}},
  title = {{Universal Tensor Squares for Symmetric Groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf}{OAI:Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026}},
  year = {2026}
}

A Cyclic Polytabloid Proof of Saxl's Conjecture

September 24, 2026 17 pages Main result formalized in Lean

For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026,
  author = {{OpenAI}},
  title = {{A Cyclic Polytabloid Proof of Saxl's Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf}{OAI:A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/205.md.

Saxl’s conjecture and universal tensor squares

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the universal tensor-square conjecture for symmetric groups in the stated range. For every positive integer $n\notin\{2,4,9\}$ , it constructs an irreducible complex representation of $S_n$ whose tensor square contains every irreducible complex representation of $S_n$ . Equivalently, all corresponding Kronecker coefficients are positive. The statement also supplies injective intertwining maps for arbitrary finite-dimensional irreducible representations.

Saxl's conjecture asserts that the tensor square of each staircase Specht module contains every irreducible representation of the corresponding symmetric group. The formalization establishes this for every $m\ge1$ : if $\rho_m=(m,m-1,\ldots,1)$ , then $g(\rho_m,\rho_m,\mu)>0$ for every partition $\mu$ of $m(m+1)/2$ .

The stronger claim that every constituent appears in the orbit span of one prescribed tensor is not included.

Comparator links

Result	Comparator statement
Universal irreducible tensor squares for symmetric groups	UniversalTensorSquares.lean
Saxl's conjecture	Saxl.lean

At a glance

Subject: Algebra
Manuscripts: 2
Pages: 54
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Saxl's conjectureProved
Tensor square conjecture for symmetric groupsProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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