The Saxl conjecture for fourth powers via the semigroup property

Abstract

The tensor square conjecture states that for n ≥ 10, there is an irreducible representation V of the symmetric group Sn such that V⊗V contains every irreducible representation of S[subscript n]. Our main result is that for large enough n, there exists an irreducible representation V such that V[superscript ⊗4] contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain (1−o(1))-fraction of irreducible representations with respect to two natural probability distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.