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)

Manuscripts (2)

Universal Tensor Squares for Symmetric Groups

September 24, 2026 · source · cite as \cite{OAI:Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026}

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.

A Cyclic Polytabloid Proof of Saxl's Conjecture

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026}

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.

Result 205 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.