Constant spectral Mackey functors and the equivariant Steenrod algebra

Abstract: For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}<em>1$-ring spectrum $R_G$. When $G=C</em>{p<sup>n}$ is a cyclic $p$-group, we provide an explicit construction of the $\mathbb{E}<em>1$-ring spectrum $R</em>{C_{p<sup>n}}$. The proof uses a reconstruction theorem for genuine $C_{p<sup>n}$-spectra, which might be of independent interest. As an application, we compute the $\mathbb{Z}$-graded $C_{p<sup>n}$-equivariant mod $p$ Steenrod algebra.