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 $H\\underline{\\mathbb{F}}_p$.