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.

arXiv comment51 pages

References in corpus (1)

• On Nonzero Kronecker Coefficients and their Consequences for Spectra