  The tensor square conjecture states that for $n \geq 10$, 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.
