A Tensor-Cube Version of the Saxl Conjecture

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

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