The Saxl Conjecture and the Dominance Order

TL;DR

This paper advances the Saxl conjecture by proving the occurrence of certain irreducible representations in tensor squares of staircase partitions, specifically those comparable in dominance order and hook partitions.

Contribution

It proves the occurrence of irreducible representations comparable to the staircase partition in dominance order and extends results to hook partitions, advancing understanding of the Saxl conjecture.

Findings

Irreducibles comparable to the staircase partition are shown to occur in tensor squares.

All irreducibles corresponding to hook partitions are shown to occur.

The results generalize previous findings by Pak, Panova, and Vallejo.

Abstract

In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook partitions. This generalizes results by Pak, Panova, and Vallejo from 2013.