Symmetric Powers and Eilenberg--Maclane Spectra

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