Steinberg Summands in the free ${\mathbb{F}}_p$-module on the Equivariant Sphere Spectrum

Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor ${\underset{‾}{\mathbb{F}}}_p$, denoted $H{\underset{‾}{\mathbb{F}}}_p$, is modeled by the free ${\mathbb{F}}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length’ filtration $\{(H{\underset{‾}{\mathbb{F}}}_p{)}_n{\}}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H{\underset{‾}{\mathbb{F}}}_p{)}_{p^k}\mathrm{/}(H{\underset{‾}{\mathbb{F}}}_p{)}_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}\mathrm{/}p{)}^k$. This is a generalization of the main result of [21]. We also show that when one smashes this filtration with $H{\underset{‾}{\mathbb{F}}}_p$, the filtration splits into its layers. The future goal of this work is to compute the $C_p$-equivariant dual Steenrod algebra $H\underline{$\mathbb{F}$}_p\wedge H\underline{$\mathbb{F}$}_p$when$p>2$, via explicit cellular constructions of equivariant classifying spaces.