Equivariant Steinberg Summands

Abstract

We construct Steinberg summands of $G$-equivariant spectra with ${\mathrm{GL}}_n({\mathbb{F}}_p)$-action. We prove a lemma about their fixed points when $G$ is a $p$-group, and then use this lemma to compute the fixed points of the Steinberg summand of the equivariant classifying space of $(\mathbb{Z}/p{)}^n$. These results will be used in a companion paper to study the layers in the mod $p$ symmetric power filtration for $H{\underline{\mathbb{F}}}_p$.

18 pages