Saxl’s conjecture and universal tensor squares

The staircase representation's tensor square contains every irreducible representation, proving Saxl's conjecture.

2 papers September 24, 2026 Explained by AI

The symmetric group shuffles n objects, and its basic building-block representations are labelled by partitions of n. Saxl conjectured in 2012 that the staircase partition's representation, multiplied by itself, contains every building block. Two papers prove this, with a Lean check. They also prove that, for every n except 2, 4 and 9, some building block's square contains all of them.

First step

2012

Jan Saxl, UCLA Combinatorics Seminar (recorded by Pak, Panova and Vallejo)

Significance

Major

It settles Saxl's conjecture and the tensor square conjecture, both well-known problems in algebraic combinatorics, with a Lean check.

Trust

Formal proof listed

Field

Algebra

2 papers

The explanation

Take the six shuffles of three objects, the symmetric group S₃. It has three irreducible representations, the building blocks that cannot be split further. They have dimensions 1, 1 and 2: the trivial one, the sign, and the standard one. Multiply the standard one by itself, a tensor square. Its dimension is 2 × 2 = 4. A character calculation shows it splits as 1 + 1 + 2. Every building block appears exactly once.

Try S₄, with 24 shuffles. Its building blocks have dimensions 1, 3, 2, 3 and 1. The only candidate with a chance is the two-dimensional one, for a reason explained below. Its square has dimension 4, and it splits as 1 + 1 + 2. Both three-dimensional blocks are missing. So for four objects no building block works.

Building blocks of the symmetric group on n objects are labelled by partitions of n: ways to write n as a decreasing sum. The standard block of S₃ is the partition 2 + 1, drawn as a staircase of rows 2 and 1. The number of times one block appears in the product of two others is a Kronecker coefficient. These numbers are notoriously hard to understand.

Saxl's conjecture is about staircases, with rows m, m − 1, down to 1. It says the square of the staircase block contains every building block. Jan Saxl proposed it at a UCLA seminar on 20 March 2012Source 1. Much partial support existed. Ikenmeyer proved it for blocks comparable with the staircase in a natural orderSource 2. Luo and Sellke showed the square contains almost all blocksSource 3.

The cyclic polytabloid paper proves Saxl's conjecture for every staircaseSource 4. The proof works inside one explicitly generated piece of the squareSource 5. The S₃ example above is the case m = 2.

The universal tensor squares paper goes further. For every n except 2, 4 and 9, some building block has a square containing every building blockSource 6 Source 7. Pak, Panova and Vallejo had stated this as the tensor square conjectureSource 8. A working block must equal its own mirror image, the partition flipped across its diagonalSource 9. That is why only the two-dimensional block of S₄ had a chance.

Staircases exist only when n is a triangle number, like 3, 6 or 10. The new task is to reach every other nSource 10. The paper enlarges a staircase by matching additions to its first two rows and columns. The one obstacle is that added boxes can create terms that cancelSource 11. Two separate tests show that the enlarged square still contains each target block. It reuses the stronger form of the staircase resultSource 12. An exact computer calculation checks every remaining size up to 64Source 13.

How sure is this? A computer proof checker called Lean has checked both main statements. It confirms the staircase result for every m. It confirms that for every n except 2, 4 and 9 some block's square contains every block. The stronger claim about one explicitly generated piece is not included. No specialist review of these papers has been published.

What it doesn't show. This does not compute the multiplicities, called Kronecker coefficients; it only shows they are all positive.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major. Lean checks both main statements; the stronger cyclic form is not checked. No specialist review is published.

Words used

Symmetric group: All the ways to shuffle n objects, combined by doing one shuffle after another.
Irreducible representation: A way for the group to act by matrices that cannot be split into smaller such actions.
Partition: A way to write n as a decreasing sum of positive whole numbers, drawn as rows of boxes.
Tensor square: A representation multiplied by itself, so the group acts on pairs at once.
Kronecker coefficient: How many times one building block appears inside the product of two others.
Staircase partition: The partition with rows m, m − 1, down to 1.
Self-conjugate: A partition that looks the same after flipping it across its diagonal.

The original papers (2)

Universal Tensor Squares for Symmetric Groups (PDF, opens in a new tab)
September 24, 2026
A Cyclic Polytabloid Proof of Saxl's Conjecture (PDF, opens in a new tab)
September 24, 2026 Formal proof listed

Lean notes

All quoted lines (13)

Saxl proposed the staircase assertion in a UCLA Combinatorics Seminar on 20 March 2012, as recorded by Pak, Panova, and Vallejo.
A Cyclic Polytabloid Proof of Saxl's Conjecture Back to text
Ikenmeyer proved positivity when $\lambda$ and $\rho_m$ are comparable in the dominance order.
A Cyclic Polytabloid Proof of Saxl's Conjecture Back to text
Luo and Sellke proved that staircase squares contain almost all partitions under both the uniform and Plancherel measures, and obtained full fourth-power coverage in all sufficiently large degrees, using modified shapes at nontriangular degrees.
Universal Tensor Squares for Symmetric Groups Back to text
For every integer $m\ge1$ and every partition $\lambda\vdash N_m$ , $g(\rho_m,\rho_m,\lambda)>0.$
A Cyclic Polytabloid Proof of Saxl's Conjecture Back to text
The proof establishes it inside one explicitly generated submodule of the tensor square, a stronger conclusion that retains the vector needed at each induction step.
A Cyclic Polytabloid Proof of Saxl's Conjecture Back to text
For every positive integer $n\notin\{2,4,9\}$ there is a partition $\lambda\vdash n$ such that $g(\lambda,\lambda,\nu)>0\qquad\text{for every }\nu\vdash n.$
Universal Tensor Squares for Symmetric Groups Back to text
Thus one irreducible representation has a tensor square containing every irreducible representation of $S_n$ .
Universal Tensor Squares for Symmetric Groups Back to text
Pak, Panova, and Vallejo formulate the assertion for $n\ge3$ , with $n\ne4,9$ , as the tensor square conjecture for symmetric groups.
Universal Tensor Squares for Symmetric Groups Back to text
The latter forces $\lambda$ to be self-conjugate, since $g(\lambda,\lambda,(1^n)) =\dim\Hom_{S_n}(S^\lambda,S^{\lambda^t}) =\begin{cases}1,&\lambda=\lambda^t,\\0,&\lambda\ne\lambda^t. \end{cases}$ Our construction meets this necessary condition throughout.
Universal Tensor Squares for Symmetric Groups Back to text
The new task here is to pass from staircase degrees to all the degrees in Theorem ….
Universal Tensor Squares for Symmetric Groups Back to text
The obstacle is not simply to place a smaller staircase inside a larger diagram.
Universal Tensor Squares for Symmetric Groups Back to text
We use the stronger cyclic form of the staircase result proved in A Cyclic Polytabloid Proof of Saxl's Conjecture.
Universal Tensor Squares for Symmetric Groups Back to text
The Murnaghan–Nakayama recursion of Section … verifies every eligible degree at most $64$ .
Universal Tensor Squares for Symmetric Groups Back to text

Other sources

I. Pak, G. Panova and E. Vallejo, “Kronecker products, characters, partitions, and the tensor square conjectures”, Advances in Mathematics 288 (2016), 702–731

Browse Algebra Named conjectures Open since the 2010s Lean-checked results