The invariant cell number property for $\mathbb{E}_4$-rings

Abstract

For any non-zero $\mathbb{E}_4$-ring $R$, we prove that an $n$-fold direct sum of suspensions of $R$ cannot admit an exhaustive filtration by fewer than $n$ cells. We do this by showing that certain integral representations of the symmetric group lift to Schur functors on additively $\mathbb{E}_3$-monoidal $\infty$-categories. We then reduce the statement to a vanishing fact for rational Schur functors due to Deligne.