A Tensor-Cube Version of the Saxl Conjecture

Let $n$ be a positive integer, and let ${\rho }_n=(n,n-1,n-2,\dots ,1)$ be the ``staircase’’ partition of size $N=\left(\genfrac{}{}{0px}{}{n+1}{2}\right)$. 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 $S^{{\rho }_n}\otimes S^{{\rho }_n}\otimes S^{{\rho }_n}$.