The Saxl conjecture for fourth powers via the semigroup property

Abstract

The tensor square conjecture states that for $$n \ge 10$$ , there is an irreducible representation V of the symmetric group $$S_n$$ such that $$V \otimes V$$ contains every irreducible representation of $$S_n$$ . Our main result is that for large enough n, there exists an irreducible representation V such that $$V^{\otimes 4}$$ 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.