A Tensor-Cube Version of the Saxl Conjecture

TL;DR

This paper proves that for the staircase partition, all irreducible representations of the symmetric group appear in the tensor cube of the corresponding Specht module, extending previous results related to the Saxl conjecture.

Contribution

It demonstrates that the tensor cube of the Specht module for the staircase partition contains all irreducible representations, advancing the understanding of the Saxl conjecture.

Findings

All irreducible representations appear in the tensor cube

Extends previous results from tensor square to tensor cube

Provides new insights into the structure of symmetric group representations

Abstract

Let $n$ be a positive integer, and let $\rho_n = (n, n-1, n-2, \ldots, 1)$ be the ``staircase'' partition of size $N = {n+1 \choose 2}$. The Saxl conjecture asserts that every irreducible representation $S^\lambda$ of the symmetric group $S_N$ appears as a subrepresentation of the tensor square $S^{\rho_n} \otimes S^{\rho_n}$. In this short note we show that every irreducible representation of $S_N$ appears in the tensor cube $S^{\rho_n} \otimes S^{\rho_n} \otimes S^{\rho_n}$.