The Saxl Conjecture for Fourth Powers via the Semigroup Property

TL;DR

This paper proves that for large enough n, there exists an irreducible representation of the symmetric group S_n whose fourth tensor power contains all irreducible representations, extending the tensor square conjecture using the semigroup property.

Contribution

It establishes the existence of a single irreducible representation whose fourth tensor power covers all irreducibles for large n, advancing the tensor square conjecture.

Findings

Existence of V with V^{ ensor 4} containing all irreducibles for large n

Tensor squares of certain irreducibles contain a large fraction of irreducibles

Application of the semigroup property to decompose partitions

Abstract

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.