Saxl's conjecture and universal tensor squares at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Saxl's conjecture and universal tensor squares

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	multiplying things	Levels:	2
Category:	Algebra	Lean version:	YES! ✔
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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.

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LEVEL 1 Universal Tensor Squares for Symmetric Groups

released 2026-09-24 | 2 theorems · 22 lemmas · 29 proofs · 16,186 words | PLAY LEVEL 1 » (pdf)

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.

LEVEL 2 A Cyclic Polytabloid Proof of Saxl's Conjecture

released 2026-09-24 | 2 theorems · 7 lemmas · 11 proofs · 7,410 words | PLAY LEVEL 2 » (pdf)

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.

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