The Saxl conjecture for fourth powers via the semigroup property

The tensor square conjecture states that for $$n \ge 10$$n≥10, there is an irreducible representation V of the symmetric group $$S_n$$Sn such that $$V \otimes V$$V⊗V contains every irreducible representation of $$S_n$$Sn. Our main result is that for large enough n, there exists an irreducible representation V such that $$V^{\otimes 4}$$V⊗4 contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain $$(1-o(1))$$(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.