Symmetric Powers and Eilenberg--Maclane Spectra

We filter the equivariant Eilenberg Maclane spectrum $H\underline{\mathbb{F}}_p$ using the mod $p$ symmetric powers of the equivariant sphere spectrum, $\mathrm{Sp}_{\mathbb{Z}/p}^{\infty}(\Sigma^{\infty G}S^0)$. When $G$ is a $p$-group, we show that the layers in the filtration are the Steinberg summands of the equivariant classifying spaces of $(\mathbb{Z}/p)^n$ for $n=0, 1, 2, \ldots$. We show that the layers of the filtration split after smashing with $H\underline{\mathbb{F}}_p$. Along the way, we produced a general computation of the geometric fixed points of $H\underline{\mathbb{Z}}$ and $H\underline{\mathbb{F}}_p$ by using symmetric powers.