Hopf ring structure on the mod $p$ cohomology of symmetric groups

Abstract

We describe a Hopf ring structure on ${\oplus }_{n\ge 0}{H}^{\ast }\left({\Sigma }_n;{\mathbb{Z}}_p\right)$, discovered by Strickland and Turner, where ${\Sigma }_n$ is the symmetric group of $n$ objects and $p$ is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For $p=2$ this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.