# Equivariant Steinberg Summands

## TL;DR

This paper constructs and analyzes Steinberg summands in $G$-equivariant spectra with $ ext{GL}_n( extbf{F}_p)$-action, providing fixed point computations crucial for understanding mod $p$ symmetric power filtrations.

## Contribution

It introduces a method to construct Steinberg summands in equivariant spectra and computes their fixed points, advancing the understanding of equivariant algebraic topology.

## Key findings

- Fixed points of Steinberg summands for $p$-groups are characterized.
- Explicit fixed point computations for the equivariant classifying space of $( extbf{Z}/p)^n$.
- Results facilitate future analysis of mod $p$ symmetric power filtrations.

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