# Saxl’s conjecture and universal tensor squares > The staircase representation's tensor square contains every irreducible representation, proving Saxl's conjecture. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-205/ Last updated: 2026-10-14 **The staircase representation's tensor square contains every irreducible representation, proving Saxl's conjecture.** **Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 24, 2026 · **Field:** Algebra · **Entry:** 205 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. 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. ## At a glance - **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 2012[1]. Much partial support existed. Ikenmeyer proved it for blocks comparable with the staircase in a natural order[2]. Luo and Sellke showed the square contains almost all blocks[3]. The cyclic polytabloid paper proves Saxl's conjecture for every staircase[4]. The proof works inside one explicitly generated piece of the square[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 block[6,7]. Pak, Panova and Vallejo had stated this as the tensor square conjecture[8]. A working block must equal its own mirror image, the partition flipped across its diagonal[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 n[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 cancel[11]. Two separate tests show that the enlarged square still contains each target block. It reuses the stronger form of the staircase result[12]. An exact computer calculation checks every remaining size up to 64[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](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf) - September 24, 2026 - [A Cyclic Polytabloid Proof of Saxl's Conjecture](https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf) - September 24, 2026 · Formal proof listed - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/205.md) ## Quoted lines from the paper (13) [1] From [A Cyclic Polytabloid Proof of Saxl's Conjecture](https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf): > Saxl proposed the staircase assertion in a UCLA Combinatorics Seminar on 20 March 2012, as recorded by Pak, Panova, and Vallejo. [2] From [A Cyclic Polytabloid Proof of Saxl's Conjecture](https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf): > Ikenmeyer proved positivity when $\lambda$ and $\rho_m$ are comparable in the dominance order. [3] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > 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. [4] From [A Cyclic Polytabloid Proof of Saxl's Conjecture](https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf): > For every integer $m\ge1$ and every partition $\lambda\vdash N_m$, \[ g(\rho_m,\rho_m,\lambda)>0. \] [5] From [A Cyclic Polytabloid Proof of Saxl's Conjecture](https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf): > 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. [6] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > For every positive integer $n\notin\{2,4,9\}$ there is a partition $\lambda\vdash n$ such that \[ g(\lambda,\lambda,\nu)>0\qquad\text{for every }\nu\vdash n. \] [7] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > Thus one irreducible representation has a tensor square containing every irreducible representation of $S_n$. [8] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > Pak, Panova, and Vallejo formulate the assertion for $n\ge3$, with $n\ne4,9$, as the \emph{tensor square conjecture} for symmetric groups. [9] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > 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. [10] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > The new task here is to pass from staircase degrees to all the degrees in Theorem …. [11] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > The obstacle is not simply to place a smaller staircase inside a larger diagram. [12] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > We use the stronger cyclic form of the staircase result proved in \emph{A Cyclic Polytabloid Proof of Saxl's Conjecture}. [13] From [Universal Tensor Squares for Symmetric Groups](https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf): > The Murnaghan--Nakayama recursion of Section … verifies every eligible degree at most $64$. ## Other sources - [I. Pak, G. Panova and E. 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